Connectivity related functions¶

This module implements the connectivity based functions for graphs and digraphs. The methods in this module are also available as part of GenericGraph, DiGraph or Graph classes as aliases, and these methods can be accessed through this module or as class methods. Here is what the module can do:

For both directed and undirected graphs:

`is_connected()`	Check whether the (di)graph is connected.
`connected_components()`	Return the list of connected components
`connected_components_number()`	Return the number of connected components.
`connected_components_subgraphs()`	Return a list of connected components as graph objects.
`connected_component_containing_vertex()`	Return a list of the vertices connected to vertex.
`connected_components_sizes()`	Return the sizes of the connected components as a list.
`blocks_and_cut_vertices()`	Return the blocks and cut vertices of the graph.
`blocks_and_cuts_tree()`	Return the blocks-and-cuts tree of the graph.
`is_cut_edge()`	Check whether the input edge is a cut-edge or a bridge.
`is_edge_cut()`	Check whether the input edges form an edge cut.
`is_cut_vertex()`	Check whether the input vertex is a cut-vertex.
`is_vertex_cut()`	Check whether the input vertices form a vertex cut.
`edge_connectivity()`	Return the edge connectivity of the graph.
`vertex_connectivity()`	Return the vertex connectivity of the graph.

For DiGraph:

`is_strongly_connected()`	Check whether the current `DiGraph` is strongly connected.
`strongly_connected_components_digraph()`	Return the digraph of the strongly connected components
`strongly_connected_components_subgraphs()`	Return the strongly connected components as a list of subgraphs.
`strongly_connected_component_containing_vertex()`	Return the strongly connected component containing a given vertex.
`strong_articulation_points()`	Return the strong articulation points of this digraph.

For undirected graphs:

`bridges()`	Return an iterator over the bridges (or cut edges) of given undirected graph.
`cleave()`	Return the connected subgraphs separated by the input vertex cut.
`is_triconnected()`	Check whether the graph is triconnected.
`spqr_tree()`	Return a SPQR-tree representing the triconnected components of the graph.
`spqr_tree_to_graph()`	Return the graph represented by the SPQR-tree \(T\).
`minimal_separators()`	Return an iterator over the minimal separators of `G`.

Methods¶

class sage.graphs.connectivity.TriconnectivitySPQR[source]¶

Bases: object

Decompose a graph into triconnected components and build SPQR-tree.

This class implements the algorithm proposed by Hopcroft and Tarjan in [Hopcroft1973], and later corrected by Gutwenger and Mutzel in [Gut2001], for finding the triconnected components of a biconnected graph. It then organizes these components into a SPQR-tree. See the:wikipedia:\(SPQR_tree\).

A SPQR-tree is a tree data structure used to represent the triconnected components of a biconnected (multi)graph and the 2-vertex cuts separating them. A node of a SPQR-tree, and the graph associated with it, can be one of the following four types:

'S' – the associated graph is a cycle with at least three vertices 'S' stands for series and is also called a polygon
'P' – the associated graph is a dipole graph, a multigraph with two vertices and three or more edges. 'P' stands for parallel and the node is called a bond.
'Q' – the associated graph has a single real edge. This trivial case is necessary to handle the graph that has only one edge.
'R' – the associated graph is a 3-vertex-connected graph that is not a cycle or dipole. 'R' stands for rigid.

The edges of the tree indicate the 2-vertex cuts of the graph.

INPUT:

G – graph; if G is a DiGraph, the computation is done on the underlying Graph (i.e., ignoring edge orientation)
check – boolean (default: True); indicates whether G needs to be tested for biconnectivity

See also

EXAMPLES:

Example from the Wikipedia article SPQR_tree:

Sage

sage: from sage.graphs.connectivity import TriconnectivitySPQR
sage: from sage.graphs.connectivity import spqr_tree_to_graph
sage: G = Graph([(1, 2), (1, 4), (1, 8), (1, 12), (3, 4), (2, 3),
....: (2, 13), (3, 13), (4, 5), (4, 7), (5, 6), (5, 8), (5, 7), (6, 7),
....: (8, 11), (8, 9), (8, 12), (9, 10), (9, 11), (9, 12), (10, 12)])
sage: tric = TriconnectivitySPQR(G)
sage: T = tric.get_spqr_tree()
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import TriconnectivitySPQR
>>> from sage.graphs.connectivity import spqr_tree_to_graph
>>> G = Graph([(Integer(1), Integer(2)), (Integer(1), Integer(4)), (Integer(1), Integer(8)), (Integer(1), Integer(12)), (Integer(3), Integer(4)), (Integer(2), Integer(3)),
... (Integer(2), Integer(13)), (Integer(3), Integer(13)), (Integer(4), Integer(5)), (Integer(4), Integer(7)), (Integer(5), Integer(6)), (Integer(5), Integer(8)), (Integer(5), Integer(7)), (Integer(6), Integer(7)),
... (Integer(8), Integer(11)), (Integer(8), Integer(9)), (Integer(8), Integer(12)), (Integer(9), Integer(10)), (Integer(9), Integer(11)), (Integer(9), Integer(12)), (Integer(10), Integer(12))])
>>> tric = TriconnectivitySPQR(G)
>>> T = tric.get_spqr_tree()
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True

An example from [Hopcroft1973]:

Sage

sage: G = Graph([(1, 2), (1, 4), (1, 8), (1, 12), (1, 13), (2, 3),
....: (2, 13), (3, 4), (3, 13), (4, 5), (4, 7), (5, 6), (5, 7), (5, 8),
....: (6, 7), (8, 9), (8, 11), (8, 12), (9, 10), (9, 11), (9, 12),
....: (10, 11), (10, 12)])
sage: tric = TriconnectivitySPQR(G)
sage: T = tric.get_spqr_tree()
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True
sage: tric.print_triconnected_components()
Triconnected: [(8, 9, None), (9, 12, None), (9, 11, None), (8, 11, None), (10, 11, None), (9, 10, None), (10, 12, None), (8, 12, 'newVEdge0')]
Bond: [(8, 12, None), (8, 12, 'newVEdge0'), (8, 12, 'newVEdge1')]
Polygon: [(6, 7, None), (5, 6, None), (7, 5, 'newVEdge2')]
Bond: [(7, 5, 'newVEdge2'), (5, 7, 'newVEdge3'), (5, 7, None)]
Polygon: [(5, 7, 'newVEdge3'), (4, 7, None), (5, 4, 'newVEdge4')]
Bond: [(5, 4, 'newVEdge4'), (4, 5, 'newVEdge5'), (4, 5, None)]
Polygon: [(4, 5, 'newVEdge5'), (5, 8, None), (1, 4, 'newVEdge9'), (1, 8, 'newVEdge10')]
Triconnected: [(1, 2, None), (2, 13, None), (1, 13, None), (3, 13, None), (2, 3, None), (1, 3, 'newVEdge7')]
Polygon: [(1, 3, 'newVEdge7'), (3, 4, None), (1, 4, 'newVEdge8')]
Bond: [(1, 4, None), (1, 4, 'newVEdge8'), (1, 4, 'newVEdge9')]
Bond: [(1, 8, None), (1, 8, 'newVEdge10'), (1, 8, 'newVEdge11')]
Polygon: [(8, 12, 'newVEdge1'), (1, 8, 'newVEdge11'), (1, 12, None)]

Python

>>> from sage.all import *
>>> G = Graph([(Integer(1), Integer(2)), (Integer(1), Integer(4)), (Integer(1), Integer(8)), (Integer(1), Integer(12)), (Integer(1), Integer(13)), (Integer(2), Integer(3)),
... (Integer(2), Integer(13)), (Integer(3), Integer(4)), (Integer(3), Integer(13)), (Integer(4), Integer(5)), (Integer(4), Integer(7)), (Integer(5), Integer(6)), (Integer(5), Integer(7)), (Integer(5), Integer(8)),
... (Integer(6), Integer(7)), (Integer(8), Integer(9)), (Integer(8), Integer(11)), (Integer(8), Integer(12)), (Integer(9), Integer(10)), (Integer(9), Integer(11)), (Integer(9), Integer(12)),
... (Integer(10), Integer(11)), (Integer(10), Integer(12))])
>>> tric = TriconnectivitySPQR(G)
>>> T = tric.get_spqr_tree()
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True
>>> tric.print_triconnected_components()
Triconnected: [(8, 9, None), (9, 12, None), (9, 11, None), (8, 11, None), (10, 11, None), (9, 10, None), (10, 12, None), (8, 12, 'newVEdge0')]
Bond: [(8, 12, None), (8, 12, 'newVEdge0'), (8, 12, 'newVEdge1')]
Polygon: [(6, 7, None), (5, 6, None), (7, 5, 'newVEdge2')]
Bond: [(7, 5, 'newVEdge2'), (5, 7, 'newVEdge3'), (5, 7, None)]
Polygon: [(5, 7, 'newVEdge3'), (4, 7, None), (5, 4, 'newVEdge4')]
Bond: [(5, 4, 'newVEdge4'), (4, 5, 'newVEdge5'), (4, 5, None)]
Polygon: [(4, 5, 'newVEdge5'), (5, 8, None), (1, 4, 'newVEdge9'), (1, 8, 'newVEdge10')]
Triconnected: [(1, 2, None), (2, 13, None), (1, 13, None), (3, 13, None), (2, 3, None), (1, 3, 'newVEdge7')]
Polygon: [(1, 3, 'newVEdge7'), (3, 4, None), (1, 4, 'newVEdge8')]
Bond: [(1, 4, None), (1, 4, 'newVEdge8'), (1, 4, 'newVEdge9')]
Bond: [(1, 8, None), (1, 8, 'newVEdge10'), (1, 8, 'newVEdge11')]
Polygon: [(8, 12, 'newVEdge1'), (1, 8, 'newVEdge11'), (1, 12, None)]

An example from [Gut2001]:

Sage

sage: G = Graph([(1, 2), (1, 4), (2, 3), (2, 5), (3, 4), (3, 5), (4, 5),
....: (4, 6), (5, 7), (5, 8), (5, 14), (6, 8), (7, 14), (8, 9), (8, 10),
....: (8, 11), (8, 12), (9, 10), (10, 13), (10, 14), (10, 15), (10, 16),
....: (11, 12), (11, 13), (12, 13), (14, 15), (14, 16), (15, 16)])
sage: T = TriconnectivitySPQR(G).get_spqr_tree()
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True

Python

>>> from sage.all import *
>>> G = Graph([(Integer(1), Integer(2)), (Integer(1), Integer(4)), (Integer(2), Integer(3)), (Integer(2), Integer(5)), (Integer(3), Integer(4)), (Integer(3), Integer(5)), (Integer(4), Integer(5)),
... (Integer(4), Integer(6)), (Integer(5), Integer(7)), (Integer(5), Integer(8)), (Integer(5), Integer(14)), (Integer(6), Integer(8)), (Integer(7), Integer(14)), (Integer(8), Integer(9)), (Integer(8), Integer(10)),
... (Integer(8), Integer(11)), (Integer(8), Integer(12)), (Integer(9), Integer(10)), (Integer(10), Integer(13)), (Integer(10), Integer(14)), (Integer(10), Integer(15)), (Integer(10), Integer(16)),
... (Integer(11), Integer(12)), (Integer(11), Integer(13)), (Integer(12), Integer(13)), (Integer(14), Integer(15)), (Integer(14), Integer(16)), (Integer(15), Integer(16))])
>>> T = TriconnectivitySPQR(G).get_spqr_tree()
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True

An example with multi-edges and accessing the triconnected components:

Sage

sage: G = Graph([(1, 2), (1, 5), (1, 5), (2, 3), (2, 3), (3, 4), (4, 5)], multiedges=True)
sage: tric = TriconnectivitySPQR(G)
sage: T = tric.get_spqr_tree()
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True
sage: tric.print_triconnected_components()
Bond:  [(1, 5, None), (1, 5, None), (1, 5, 'newVEdge0')]
Bond:  [(2, 3, None), (2, 3, None), (2, 3, 'newVEdge1')]
Polygon:  [(4, 5, None), (1, 5, 'newVEdge0'), (3, 4, None), (2, 3, 'newVEdge1'), (1, 2, None)]

Python

>>> from sage.all import *
>>> G = Graph([(Integer(1), Integer(2)), (Integer(1), Integer(5)), (Integer(1), Integer(5)), (Integer(2), Integer(3)), (Integer(2), Integer(3)), (Integer(3), Integer(4)), (Integer(4), Integer(5))], multiedges=True)
>>> tric = TriconnectivitySPQR(G)
>>> T = tric.get_spqr_tree()
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True
>>> tric.print_triconnected_components()
Bond:  [(1, 5, None), (1, 5, None), (1, 5, 'newVEdge0')]
Bond:  [(2, 3, None), (2, 3, None), (2, 3, 'newVEdge1')]
Polygon:  [(4, 5, None), (1, 5, 'newVEdge0'), (3, 4, None), (2, 3, 'newVEdge1'), (1, 2, None)]

An example of a triconnected graph:

Sage

sage: G = Graph([('a', 'b'), ('a', 'c'), ('a', 'd'), ('b', 'c'), ('b', 'd'), ('c', 'd')])
sage: T = TriconnectivitySPQR(G).get_spqr_tree()
sage: print(T.vertices(sort=True))
[('R', Multi-graph on 4 vertices)]
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True

Python

>>> from sage.all import *
>>> G = Graph([('a', 'b'), ('a', 'c'), ('a', 'd'), ('b', 'c'), ('b', 'd'), ('c', 'd')])
>>> T = TriconnectivitySPQR(G).get_spqr_tree()
>>> print(T.vertices(sort=True))
[('R', Multi-graph on 4 vertices)]
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True

An example of a directed graph with multi-edges:

Sage

sage: G = DiGraph([(1, 2), (2, 3), (3, 4), (4, 5), (1, 5), (5, 1)])
sage: tric = TriconnectivitySPQR(G)
sage: tric.print_triconnected_components()
Bond:  [(1, 5, None), (5, 1, None), (1, 5, 'newVEdge0')]
Polygon:  [(4, 5, None), (1, 5, 'newVEdge0'), (3, 4, None), (2, 3, None), (1, 2, None)]

Python

>>> from sage.all import *
>>> G = DiGraph([(Integer(1), Integer(2)), (Integer(2), Integer(3)), (Integer(3), Integer(4)), (Integer(4), Integer(5)), (Integer(1), Integer(5)), (Integer(5), Integer(1))])
>>> tric = TriconnectivitySPQR(G)
>>> tric.print_triconnected_components()
Bond:  [(1, 5, None), (5, 1, None), (1, 5, 'newVEdge0')]
Polygon:  [(4, 5, None), (1, 5, 'newVEdge0'), (3, 4, None), (2, 3, None), (1, 2, None)]

Edge labels are preserved by the construction:

Sage

sage: G = Graph([(0, 1, '01'), (0, 4, '04'), (1, 2, '12'), (1, 5, '15'),
....: (2, 3, '23'), (2, 6, '26'), (3, 7, '37'), (4, 5, '45'),
....: (5, 6, '56'), (6, 7, 67)])
sage: T = TriconnectivitySPQR(G).get_spqr_tree()
sage: H = spqr_tree_to_graph(T)
sage: all(G.has_edge(e) for e in H.edge_iterator())
True
sage: all(H.has_edge(e) for e in G.edge_iterator())
True

Python

>>> from sage.all import *
>>> G = Graph([(Integer(0), Integer(1), '01'), (Integer(0), Integer(4), '04'), (Integer(1), Integer(2), '12'), (Integer(1), Integer(5), '15'),
... (Integer(2), Integer(3), '23'), (Integer(2), Integer(6), '26'), (Integer(3), Integer(7), '37'), (Integer(4), Integer(5), '45'),
... (Integer(5), Integer(6), '56'), (Integer(6), Integer(7), Integer(67))])
>>> T = TriconnectivitySPQR(G).get_spqr_tree()
>>> H = spqr_tree_to_graph(T)
>>> all(G.has_edge(e) for e in H.edge_iterator())
True
>>> all(H.has_edge(e) for e in G.edge_iterator())
True

get_spqr_tree()[source]¶

Return an SPQR-tree representing the triconnected components of the graph.

An SPQR-tree is a tree data structure used to represent the triconnected components of a biconnected (multi)graph and the 2-vertex cuts separating them. A node of a SPQR-tree, and the graph associated with it, can be one of the following four types:

'S' – the associated graph is a cycle with at least three vertices. 'S' stands for series.
'P' – the associated graph is a dipole graph, a multigraph with two vertices and three or more edges. 'P' stands for parallel.
'Q' – the associated graph has a single real edge. This trivial case is necessary to handle the graph that has only one edge.
'R' – the associated graph is a 3-connected graph that is not a cycle or dipole. 'R' stands for rigid.

The edges of the tree indicate the 2-vertex cuts of the graph.

OUTPUT:

SPQR-tree a tree whose vertices are labeled with the block’s type and the subgraph of three-blocks in the decomposition.

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import TriconnectivitySPQR
sage: G = Graph(2)
sage: for i in range(3):
....:     G.add_clique([0, 1, G.add_vertex(), G.add_vertex()])
sage: tric = TriconnectivitySPQR(G)
sage: Tree = tric.get_spqr_tree()
sage: K4 = graphs.CompleteGraph(4)
sage: all(u[1].is_isomorphic(K4) for u in Tree if u[0] == 'R')
True
sage: from sage.graphs.connectivity import spqr_tree_to_graph
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G = Graph(2)
sage: for i in range(3):
....:     G.add_path([0, G.add_vertex(), G.add_vertex(), 1])
sage: tric = TriconnectivitySPQR(G)
sage: Tree = tric.get_spqr_tree()
sage: C4 = graphs.CycleGraph(4)
sage: all(u[1].is_isomorphic(C4) for u in Tree if u[0] == 'S')
True
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G.allow_multiple_edges(True)
sage: G.add_edges(G.edge_iterator())
sage: tric = TriconnectivitySPQR(G)
sage: Tree = tric.get_spqr_tree()
sage: all(u[1].is_isomorphic(C4) for u in Tree if u[0] == 'S')
True
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G = graphs.CycleGraph(6)
sage: tric = TriconnectivitySPQR(G)
sage: Tree = tric.get_spqr_tree()
sage: Tree.order()
1
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True
sage: G.add_edge(0, 3)
sage: tric = TriconnectivitySPQR(G)
sage: Tree = tric.get_spqr_tree()
sage: Tree.order()
3
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G = Graph([(0, 1)], multiedges=True)
sage: tric = TriconnectivitySPQR(G)
sage: Tree = tric.get_spqr_tree()
sage: Tree.vertices(sort=True)
[('Q', Multi-graph on 2 vertices)]
sage: G.add_edge(0, 1)
sage: Tree = TriconnectivitySPQR(G).get_spqr_tree()
sage: Tree.vertices(sort=True)
[('P', Multi-graph on 2 vertices)]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import TriconnectivitySPQR
>>> G = Graph(Integer(2))
>>> for i in range(Integer(3)):
...     G.add_clique([Integer(0), Integer(1), G.add_vertex(), G.add_vertex()])
>>> tric = TriconnectivitySPQR(G)
>>> Tree = tric.get_spqr_tree()
>>> K4 = graphs.CompleteGraph(Integer(4))
>>> all(u[Integer(1)].is_isomorphic(K4) for u in Tree if u[Integer(0)] == 'R')
True
>>> from sage.graphs.connectivity import spqr_tree_to_graph
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G = Graph(Integer(2))
>>> for i in range(Integer(3)):
...     G.add_path([Integer(0), G.add_vertex(), G.add_vertex(), Integer(1)])
>>> tric = TriconnectivitySPQR(G)
>>> Tree = tric.get_spqr_tree()
>>> C4 = graphs.CycleGraph(Integer(4))
>>> all(u[Integer(1)].is_isomorphic(C4) for u in Tree if u[Integer(0)] == 'S')
True
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G.allow_multiple_edges(True)
>>> G.add_edges(G.edge_iterator())
>>> tric = TriconnectivitySPQR(G)
>>> Tree = tric.get_spqr_tree()
>>> all(u[Integer(1)].is_isomorphic(C4) for u in Tree if u[Integer(0)] == 'S')
True
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G = graphs.CycleGraph(Integer(6))
>>> tric = TriconnectivitySPQR(G)
>>> Tree = tric.get_spqr_tree()
>>> Tree.order()
1
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True
>>> G.add_edge(Integer(0), Integer(3))
>>> tric = TriconnectivitySPQR(G)
>>> Tree = tric.get_spqr_tree()
>>> Tree.order()
3
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G = Graph([(Integer(0), Integer(1))], multiedges=True)
>>> tric = TriconnectivitySPQR(G)
>>> Tree = tric.get_spqr_tree()
>>> Tree.vertices(sort=True)
[('Q', Multi-graph on 2 vertices)]
>>> G.add_edge(Integer(0), Integer(1))
>>> Tree = TriconnectivitySPQR(G).get_spqr_tree()
>>> Tree.vertices(sort=True)
[('P', Multi-graph on 2 vertices)]

get_triconnected_components()[source]¶

Return the triconnected components as a list of tuples.

Each component is represented as a tuple of the type of the component and the list of edges of the component.

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import TriconnectivitySPQR
sage: G = Graph(2)
sage: for i in range(3):
....:     G.add_path([0, G.add_vertex(), G.add_vertex(), 1])
sage: tric = TriconnectivitySPQR(G)
sage: tric.get_triconnected_components()
[('Polygon', [(4, 5, None), (0, 4, None), (1, 5, None), (1, 0, 'newVEdge1')]),
('Polygon', [(6, 7, None), (0, 6, None), (1, 7, None), (1, 0, 'newVEdge3')]),
('Bond', [(1, 0, 'newVEdge1'), (1, 0, 'newVEdge3'), (1, 0, 'newVEdge4')]),
('Polygon', [(1, 3, None), (1, 0, 'newVEdge4'), (2, 3, None), (0, 2, None)])]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import TriconnectivitySPQR
>>> G = Graph(Integer(2))
>>> for i in range(Integer(3)):
...     G.add_path([Integer(0), G.add_vertex(), G.add_vertex(), Integer(1)])
>>> tric = TriconnectivitySPQR(G)
>>> tric.get_triconnected_components()
[('Polygon', [(4, 5, None), (0, 4, None), (1, 5, None), (1, 0, 'newVEdge1')]),
('Polygon', [(6, 7, None), (0, 6, None), (1, 7, None), (1, 0, 'newVEdge3')]),
('Bond', [(1, 0, 'newVEdge1'), (1, 0, 'newVEdge3'), (1, 0, 'newVEdge4')]),
('Polygon', [(1, 3, None), (1, 0, 'newVEdge4'), (2, 3, None), (0, 2, None)])]

print_triconnected_components()[source]¶

Print the type and list of edges of each component.

EXAMPLES:

An example from [Hopcroft1973]:

Sage

sage: from sage.graphs.connectivity import TriconnectivitySPQR
sage: from sage.graphs.connectivity import spqr_tree_to_graph
sage: G = Graph([(1, 2), (1, 4), (1, 8), (1, 12), (1, 13), (2, 3),
....: (2, 13), (3, 4), (3, 13), (4, 5), (4, 7), (5, 6), (5, 7), (5, 8),
....: (6, 7), (8, 9), (8, 11), (8, 12), (9, 10), (9, 11), (9, 12),
....: (10, 11), (10, 12)])
sage: tric = TriconnectivitySPQR(G)
sage: T = tric.get_spqr_tree()
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True
sage: tric.print_triconnected_components()
Triconnected: [(8, 9, None), (9, 12, None), (9, 11, None), (8, 11, None), (10, 11, None), (9, 10, None), (10, 12, None), (8, 12, 'newVEdge0')]
Bond: [(8, 12, None), (8, 12, 'newVEdge0'), (8, 12, 'newVEdge1')]
Polygon: [(6, 7, None), (5, 6, None), (7, 5, 'newVEdge2')]
Bond: [(7, 5, 'newVEdge2'), (5, 7, 'newVEdge3'), (5, 7, None)]
Polygon: [(5, 7, 'newVEdge3'), (4, 7, None), (5, 4, 'newVEdge4')]
Bond: [(5, 4, 'newVEdge4'), (4, 5, 'newVEdge5'), (4, 5, None)]
Polygon: [(4, 5, 'newVEdge5'), (5, 8, None), (1, 4, 'newVEdge9'), (1, 8, 'newVEdge10')]
Triconnected: [(1, 2, None), (2, 13, None), (1, 13, None), (3, 13, None), (2, 3, None), (1, 3, 'newVEdge7')]
Polygon: [(1, 3, 'newVEdge7'), (3, 4, None), (1, 4, 'newVEdge8')]
Bond: [(1, 4, None), (1, 4, 'newVEdge8'), (1, 4, 'newVEdge9')]
Bond: [(1, 8, None), (1, 8, 'newVEdge10'), (1, 8, 'newVEdge11')]
Polygon: [(8, 12, 'newVEdge1'), (1, 8, 'newVEdge11'), (1, 12, None)]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import TriconnectivitySPQR
>>> from sage.graphs.connectivity import spqr_tree_to_graph
>>> G = Graph([(Integer(1), Integer(2)), (Integer(1), Integer(4)), (Integer(1), Integer(8)), (Integer(1), Integer(12)), (Integer(1), Integer(13)), (Integer(2), Integer(3)),
... (Integer(2), Integer(13)), (Integer(3), Integer(4)), (Integer(3), Integer(13)), (Integer(4), Integer(5)), (Integer(4), Integer(7)), (Integer(5), Integer(6)), (Integer(5), Integer(7)), (Integer(5), Integer(8)),
... (Integer(6), Integer(7)), (Integer(8), Integer(9)), (Integer(8), Integer(11)), (Integer(8), Integer(12)), (Integer(9), Integer(10)), (Integer(9), Integer(11)), (Integer(9), Integer(12)),
... (Integer(10), Integer(11)), (Integer(10), Integer(12))])
>>> tric = TriconnectivitySPQR(G)
>>> T = tric.get_spqr_tree()
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True
>>> tric.print_triconnected_components()
Triconnected: [(8, 9, None), (9, 12, None), (9, 11, None), (8, 11, None), (10, 11, None), (9, 10, None), (10, 12, None), (8, 12, 'newVEdge0')]
Bond: [(8, 12, None), (8, 12, 'newVEdge0'), (8, 12, 'newVEdge1')]
Polygon: [(6, 7, None), (5, 6, None), (7, 5, 'newVEdge2')]
Bond: [(7, 5, 'newVEdge2'), (5, 7, 'newVEdge3'), (5, 7, None)]
Polygon: [(5, 7, 'newVEdge3'), (4, 7, None), (5, 4, 'newVEdge4')]
Bond: [(5, 4, 'newVEdge4'), (4, 5, 'newVEdge5'), (4, 5, None)]
Polygon: [(4, 5, 'newVEdge5'), (5, 8, None), (1, 4, 'newVEdge9'), (1, 8, 'newVEdge10')]
Triconnected: [(1, 2, None), (2, 13, None), (1, 13, None), (3, 13, None), (2, 3, None), (1, 3, 'newVEdge7')]
Polygon: [(1, 3, 'newVEdge7'), (3, 4, None), (1, 4, 'newVEdge8')]
Bond: [(1, 4, None), (1, 4, 'newVEdge8'), (1, 4, 'newVEdge9')]
Bond: [(1, 8, None), (1, 8, 'newVEdge10'), (1, 8, 'newVEdge11')]
Polygon: [(8, 12, 'newVEdge1'), (1, 8, 'newVEdge11'), (1, 12, None)]

sage.graphs.connectivity.blocks_and_cut_vertices(G, algorithm='Tarjan_Boost', sort=False, key=None)[source]¶

Return the blocks and cut vertices of the graph.

In the case of a digraph, this computation is done on the underlying graph.

A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.

INPUT:

algorithm – string (default: 'Tarjan_Boost'); the algorithm to use among:
- 'Tarjan_Boost' – default; Tarjan’s algorithm (Boost implementation)
- 'Tarjan_Sage' – Tarjan’s algorithm (Sage implementation)
sort – boolean (default: False); whether to sort vertices inside the components and the list of cut vertices currently only available for ``’Tarjan_Sage’``
key – a function (default: None); a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must have sort=True)

OUTPUT: (B, C), where B is a list of blocks - each is a list of vertices and the blocks are the corresponding induced subgraphs - and C is a list of cut vertices.

ALGORITHM:

We implement the algorithm proposed by Tarjan in [Tarjan72]. The original version is recursive. We emulate the recursion using a stack.

See also

EXAMPLES:

We construct a trivial example of a graph with one cut vertex:

Sage

sage: from sage.graphs.connectivity import blocks_and_cut_vertices
sage: rings = graphs.CycleGraph(10)
sage: rings.merge_vertices([0, 5])
sage: blocks_and_cut_vertices(rings)
([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])
sage: rings.blocks_and_cut_vertices()
([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])
sage: B, C = blocks_and_cut_vertices(rings, algorithm='Tarjan_Sage', sort=True)
sage: B, C
([[0, 1, 2, 3, 4], [0, 6, 7, 8, 9]], [0])
sage: B2, C2 = blocks_and_cut_vertices(rings, algorithm='Tarjan_Sage', sort=False)
sage: Set(map(Set, B)) == Set(map(Set, B2)) and set(C) == set(C2)
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import blocks_and_cut_vertices
>>> rings = graphs.CycleGraph(Integer(10))
>>> rings.merge_vertices([Integer(0), Integer(5)])
>>> blocks_and_cut_vertices(rings)
([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])
>>> rings.blocks_and_cut_vertices()
([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])
>>> B, C = blocks_and_cut_vertices(rings, algorithm='Tarjan_Sage', sort=True)
>>> B, C
([[0, 1, 2, 3, 4], [0, 6, 7, 8, 9]], [0])
>>> B2, C2 = blocks_and_cut_vertices(rings, algorithm='Tarjan_Sage', sort=False)
>>> Set(map(Set, B)) == Set(map(Set, B2)) and set(C) == set(C2)
True

The Petersen graph is biconnected, hence has no cut vertices:

Sage

sage: blocks_and_cut_vertices(graphs.PetersenGraph())
([[0, 1, 4, 5, 2, 6, 3, 7, 8, 9]], [])

Python

>>> from sage.all import *
>>> blocks_and_cut_vertices(graphs.PetersenGraph())
([[0, 1, 4, 5, 2, 6, 3, 7, 8, 9]], [])

Decomposing paths to pairs:

Sage

sage: g = graphs.PathGraph(4) + graphs.PathGraph(5)
sage: blocks_and_cut_vertices(g)
([[2, 3], [1, 2], [0, 1], [7, 8], [6, 7], [5, 6], [4, 5]], [1, 2, 5, 6, 7])

Python

>>> from sage.all import *
>>> g = graphs.PathGraph(Integer(4)) + graphs.PathGraph(Integer(5))
>>> blocks_and_cut_vertices(g)
([[2, 3], [1, 2], [0, 1], [7, 8], [6, 7], [5, 6], [4, 5]], [1, 2, 5, 6, 7])

A disconnected graph:

Sage

sage: g = Graph({1: {2: 28, 3: 10}, 2: {1: 10, 3: 16}, 4: {}, 5: {6: 3, 7: 10, 8: 4}})
sage: blocks_and_cut_vertices(g)
([[1, 2, 3], [5, 6], [5, 7], [5, 8], [4]], [5])

Python

>>> from sage.all import *
>>> g = Graph({Integer(1): {Integer(2): Integer(28), Integer(3): Integer(10)}, Integer(2): {Integer(1): Integer(10), Integer(3): Integer(16)}, Integer(4): {}, Integer(5): {Integer(6): Integer(3), Integer(7): Integer(10), Integer(8): Integer(4)}})
>>> blocks_and_cut_vertices(g)
([[1, 2, 3], [5, 6], [5, 7], [5, 8], [4]], [5])

A directed graph with Boost’s algorithm (Issue #25994):

Sage

sage: rings = graphs.CycleGraph(10)
sage: rings.merge_vertices([0, 5])
sage: rings = rings.to_directed()
sage: blocks_and_cut_vertices(rings, algorithm='Tarjan_Boost')
([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])

Python

>>> from sage.all import *
>>> rings = graphs.CycleGraph(Integer(10))
>>> rings.merge_vertices([Integer(0), Integer(5)])
>>> rings = rings.to_directed()
>>> blocks_and_cut_vertices(rings, algorithm='Tarjan_Boost')
([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])

sage.graphs.connectivity.blocks_and_cuts_tree(G)[source]¶

Return the blocks-and-cuts tree of self.

This new graph has two different kinds of vertices, some representing the blocks (type B) and some other the cut vertices of the graph (type C).

There is an edge between a vertex \(u\) of type B and a vertex \(v\) of type C if the cut-vertex corresponding to \(v\) is in the block corresponding to \(u\).

The resulting graph is a tree, with the additional characteristic property that the distance between two leaves is even. When self is not connected, the resulting graph is a forest.

When self is biconnected, the tree is reduced to a single node of type \(B\).

We referred to [HarPri] and [Gallai] for blocks and cuts tree.

See also

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import blocks_and_cuts_tree
sage: T = blocks_and_cuts_tree(graphs.KrackhardtKiteGraph()); T
Graph on 5 vertices
sage: T.is_isomorphic(graphs.PathGraph(5))
True
sage: from sage.graphs.connectivity import blocks_and_cuts_tree
sage: T = graphs.KrackhardtKiteGraph().blocks_and_cuts_tree(); T
Graph on 5 vertices

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import blocks_and_cuts_tree
>>> T = blocks_and_cuts_tree(graphs.KrackhardtKiteGraph()); T
Graph on 5 vertices
>>> T.is_isomorphic(graphs.PathGraph(Integer(5)))
True
>>> from sage.graphs.connectivity import blocks_and_cuts_tree
>>> T = graphs.KrackhardtKiteGraph().blocks_and_cuts_tree(); T
Graph on 5 vertices

The distance between two leaves is even:

Sage

sage: T = blocks_and_cuts_tree(graphs.RandomTree(40))
sage: T.is_tree()
True
sage: leaves = [v for v in T if T.degree(v) == 1]
sage: all(T.distance(u,v) % 2 == 0 for u in leaves for v in leaves)
True

Python

>>> from sage.all import *
>>> T = blocks_and_cuts_tree(graphs.RandomTree(Integer(40)))
>>> T.is_tree()
True
>>> leaves = [v for v in T if T.degree(v) == Integer(1)]
>>> all(T.distance(u,v) % Integer(2) == Integer(0) for u in leaves for v in leaves)
True

The tree of a biconnected graph has a single vertex, of type \(B\):

Sage

sage: T = blocks_and_cuts_tree(graphs.PetersenGraph())
sage: T.vertices(sort=True)
[('B', (0, 1, 4, 5, 2, 6, 3, 7, 8, 9))]

Python

>>> from sage.all import *
>>> T = blocks_and_cuts_tree(graphs.PetersenGraph())
>>> T.vertices(sort=True)
[('B', (0, 1, 4, 5, 2, 6, 3, 7, 8, 9))]

sage.graphs.connectivity.bridges(G, labels=True)[source]¶

Return an iterator over the bridges (or cut edges).

A bridge is an edge whose deletion disconnects the undirected graph. A disconnected graph has no bridge.

INPUT:

labels – boolean (default: True); if False, each bridge is a tuple \((u, v)\) of vertices

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import bridges
sage: from sage.graphs.connectivity import is_connected
sage: g = 2 * graphs.PetersenGraph()
sage: g.add_edge(1, 10)
sage: is_connected(g)
True
sage: list(bridges(g))
[(1, 10, None)]
sage: list(g.bridges())
[(1, 10, None)]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import bridges
>>> from sage.graphs.connectivity import is_connected
>>> g = Integer(2) * graphs.PetersenGraph()
>>> g.add_edge(Integer(1), Integer(10))
>>> is_connected(g)
True
>>> list(bridges(g))
[(1, 10, None)]
>>> list(g.bridges())
[(1, 10, None)]

Every edge of a tree is a bridge:

Sage

sage: g = graphs.RandomTree(100)
sage: sum(1 for _ in g.bridges()) == 99
True

Python

>>> from sage.all import *
>>> g = graphs.RandomTree(Integer(100))
>>> sum(Integer(1) for _ in g.bridges()) == Integer(99)
True

sage.graphs.connectivity.cleave(G, cut_vertices=None, virtual_edges=True, solver=None, verbose=0, integrality_tolerance=0.001)[source]¶

Return the connected subgraphs separated by the input vertex cut.

Given a connected (multi)graph \(G\) and a vertex cut \(X\), this method computes the list of subgraphs of \(G\) induced by each connected component \(c\) of \(G\setminus X\) plus \(X\), i.e., \(G[c\cup X]\).

INPUT:

G – a Graph
cut_vertices – iterable container of vertices (default: None); a set of vertices representing a vertex cut of G. If no vertex cut is given, the method will compute one via a call to vertex_connectivity().
virtual_edges – boolean (default: True); whether to add virtual edges to the sides of the cut or not. A virtual edge is an edge between a pair of vertices of the cut that are not connected by an edge in G.
solver – string (default: None); specifies a Mixed Integer Linear Programming (MILP) solver to be used. If set to None, the default one is used. For more information on MILP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
verbose – integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.
integrality_tolerance – float; parameter for use with MILP solvers over an inexact base ring; see MixedIntegerLinearProgram.get_values().

OUTPUT: a triple \((S, C, f)\), where

\(S\) is a list of the graphs that are sides of the vertex cut.
\(C\) is the graph of the cocycles. For each pair of vertices of the cut, if there exists an edge between them, \(C\) has one copy of each edge connecting them in G per sides of the cut plus one extra copy. Furthermore, when virtual_edges == True, if a pair of vertices of the cut is not connected by an edge in G, then it has one virtual edge between them per sides of the cut.
\(f\) is the complement of the subgraph of G induced by the vertex cut. Hence, its vertex set is the vertex cut, and its edge set is the set of virtual edges (i.e., edges between pairs of vertices of the cut that are not connected by an edge in G). When virtual_edges == False, the edge set is empty.

EXAMPLES:

If there is an edge between cut vertices:

Sage

sage: from sage.graphs.connectivity import cleave
sage: G = Graph(2)
sage: for _ in range(3):
....:     G.add_clique([0, 1, G.add_vertex(), G.add_vertex()])
sage: S1,C1,f1 = cleave(G, cut_vertices=[0, 1])
sage: [g.order() for g in S1]
[4, 4, 4]
sage: C1.order(), C1.size()
(2, 4)
sage: f1.vertices(sort=True), f1.edges(sort=True)
([0, 1], [])

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import cleave
>>> G = Graph(Integer(2))
>>> for _ in range(Integer(3)):
...     G.add_clique([Integer(0), Integer(1), G.add_vertex(), G.add_vertex()])
>>> S1,C1,f1 = cleave(G, cut_vertices=[Integer(0), Integer(1)])
>>> [g.order() for g in S1]
[4, 4, 4]
>>> C1.order(), C1.size()
(2, 4)
>>> f1.vertices(sort=True), f1.edges(sort=True)
([0, 1], [])

If virtual_edges == False and there is an edge between cut vertices:

Sage

sage: G.subgraph([0, 1]).complement() == Graph([[0, 1], []])
True
sage: S2,C2,f2 = cleave(G, cut_vertices=[0, 1], virtual_edges=False)
sage: (S1 == S2, C1 == C2, f1 == f2)
(True, True, True)

Python

>>> from sage.all import *
>>> G.subgraph([Integer(0), Integer(1)]).complement() == Graph([[Integer(0), Integer(1)], []])
True
>>> S2,C2,f2 = cleave(G, cut_vertices=[Integer(0), Integer(1)], virtual_edges=False)
>>> (S1 == S2, C1 == C2, f1 == f2)
(True, True, True)

If cut vertices doesn’t have edge between them:

Sage

sage: G.delete_edge(0, 1)
sage: S1,C1,f1 = cleave(G, cut_vertices=[0, 1])
sage: [g.order() for g in S1]
[4, 4, 4]
sage: C1.order(), C1.size()
(2, 3)
sage: f1.vertices(sort=True), f1.edges(sort=True)
([0, 1], [(0, 1, None)])

Python

>>> from sage.all import *
>>> G.delete_edge(Integer(0), Integer(1))
>>> S1,C1,f1 = cleave(G, cut_vertices=[Integer(0), Integer(1)])
>>> [g.order() for g in S1]
[4, 4, 4]
>>> C1.order(), C1.size()
(2, 3)
>>> f1.vertices(sort=True), f1.edges(sort=True)
([0, 1], [(0, 1, None)])

If virtual_edges == False and the cut vertices are not connected by an edge:

Sage

sage: G.subgraph([0, 1]).complement() == Graph([[0, 1], []])
False
sage: S2,C2,f2 = cleave(G, cut_vertices=[0, 1], virtual_edges=False)
sage: [g.order() for g in S2]
[4, 4, 4]
sage: C2.order(), C2.size()
(2, 0)
sage: f2.vertices(sort=True), f2.edges(sort=True)
([0, 1], [])
sage: (S1 == S2, C1 == C2, f1 == f2)
(False, False, False)

Python

>>> from sage.all import *
>>> G.subgraph([Integer(0), Integer(1)]).complement() == Graph([[Integer(0), Integer(1)], []])
False
>>> S2,C2,f2 = cleave(G, cut_vertices=[Integer(0), Integer(1)], virtual_edges=False)
>>> [g.order() for g in S2]
[4, 4, 4]
>>> C2.order(), C2.size()
(2, 0)
>>> f2.vertices(sort=True), f2.edges(sort=True)
([0, 1], [])
>>> (S1 == S2, C1 == C2, f1 == f2)
(False, False, False)

If \(G\) is a biconnected multigraph:

Sage

sage: G = graphs.CompleteBipartiteGraph(2, 3)
sage: G.add_edge(2, 3)
sage: G.allow_multiple_edges(True)
sage: G.add_edges(G.edge_iterator())
sage: G.add_edges([(0, 1), (0, 1), (0, 1)])
sage: S,C,f = cleave(G, cut_vertices=[0, 1])
sage: for g in S:
....:     print(g.edges(sort=True, labels=0))
[(0, 1), (0, 1), (0, 1), (0, 2), (0, 2), (0, 3), (0, 3), (1, 2), (1, 2), (1, 3), (1, 3), (2, 3), (2, 3)]
[(0, 1), (0, 1), (0, 1), (0, 4), (0, 4), (1, 4), (1, 4)]

Python

>>> from sage.all import *
>>> G = graphs.CompleteBipartiteGraph(Integer(2), Integer(3))
>>> G.add_edge(Integer(2), Integer(3))
>>> G.allow_multiple_edges(True)
>>> G.add_edges(G.edge_iterator())
>>> G.add_edges([(Integer(0), Integer(1)), (Integer(0), Integer(1)), (Integer(0), Integer(1))])
>>> S,C,f = cleave(G, cut_vertices=[Integer(0), Integer(1)])
>>> for g in S:
...     print(g.edges(sort=True, labels=Integer(0)))
[(0, 1), (0, 1), (0, 1), (0, 2), (0, 2), (0, 3), (0, 3), (1, 2), (1, 2), (1, 3), (1, 3), (2, 3), (2, 3)]
[(0, 1), (0, 1), (0, 1), (0, 4), (0, 4), (1, 4), (1, 4)]

sage.graphs.connectivity.connected_component_containing_vertex(G, vertex, sort=None, key=None, forbidden_vertices=None)[source]¶

Return a list of the vertices connected to vertex.

INPUT:

G – the input graph
vertex – the vertex to search for
sort – boolean (default: None); if True, vertices inside the component are sorted according to the default ordering

As of Issue #35889, this argument must be explicitly specified (unless a key is given); otherwise a warning is printed and sort=True is used. The default will eventually be changed to False.
key – a function (default: None); a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must have sort=True)
forbidden_vertices – list (default: None); set of vertices to avoid during the search. The start vertex cannot be in this set.

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import connected_component_containing_vertex
sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: connected_component_containing_vertex(G, 0, sort=True)
[0, 1, 2, 3]
sage: G.connected_component_containing_vertex(0, sort=True)
[0, 1, 2, 3]
sage: G.connected_component_containing_vertex(0, sort=True, forbidden_vertices=[1, 3])
[0]
sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: connected_component_containing_vertex(D, 0, sort=True)
[0, 1, 2, 3]
sage: connected_component_containing_vertex(D, 0, sort=True, key=lambda x: -x)
[3, 2, 1, 0]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import connected_component_containing_vertex
>>> G = Graph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> connected_component_containing_vertex(G, Integer(0), sort=True)
[0, 1, 2, 3]
>>> G.connected_component_containing_vertex(Integer(0), sort=True)
[0, 1, 2, 3]
>>> G.connected_component_containing_vertex(Integer(0), sort=True, forbidden_vertices=[Integer(1), Integer(3)])
[0]
>>> D = DiGraph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> connected_component_containing_vertex(D, Integer(0), sort=True)
[0, 1, 2, 3]
>>> connected_component_containing_vertex(D, Integer(0), sort=True, key=lambda x: -x)
[3, 2, 1, 0]

sage.graphs.connectivity.connected_components(G, sort=None, key=None, forbidden_vertices=None)[source]¶

Return the list of connected components.

This returns a list of lists of vertices, each list representing a connected component. The list is ordered from largest to smallest component.

INPUT:

G – the input graph
sort – boolean (default: None); if True, vertices inside each component are sorted according to the default ordering

As of Issue #35889, this argument must be explicitly specified (unless a key is given); otherwise a warning is printed and sort=True is used. The default will eventually be changed to False.
key – a function (default: None); a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must have sort=True)
forbidden_vertices – list (default: None); set of vertices to avoid during the search

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import connected_components
sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: connected_components(G, sort=True)
[[0, 1, 2, 3], [4, 5, 6]]
sage: G.connected_components(sort=True)
[[0, 1, 2, 3], [4, 5, 6]]
sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: connected_components(D, sort=True)
[[0, 1, 2, 3], [4, 5, 6]]
sage: connected_components(D, sort=True, key=lambda x: -x)
[[3, 2, 1, 0], [6, 5, 4]]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import connected_components
>>> G = Graph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> connected_components(G, sort=True)
[[0, 1, 2, 3], [4, 5, 6]]
>>> G.connected_components(sort=True)
[[0, 1, 2, 3], [4, 5, 6]]
>>> D = DiGraph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> connected_components(D, sort=True)
[[0, 1, 2, 3], [4, 5, 6]]
>>> connected_components(D, sort=True, key=lambda x: -x)
[[3, 2, 1, 0], [6, 5, 4]]

Connected components in a graph with forbidden vertices:

Sage

sage: G = graphs.PathGraph(5)
sage: connected_components(G, sort=True, forbidden_vertices=[2])
[[0, 1], [3, 4]]
sage: connected_components(G, sort=True,
....:     forbidden_vertices=G.neighbor_iterator(2, closed=True))
[[0], [4]]

Python

>>> from sage.all import *
>>> G = graphs.PathGraph(Integer(5))
>>> connected_components(G, sort=True, forbidden_vertices=[Integer(2)])
[[0, 1], [3, 4]]
>>> connected_components(G, sort=True,
...     forbidden_vertices=G.neighbor_iterator(Integer(2), closed=True))
[[0], [4]]

sage.graphs.connectivity.connected_components_number(G, forbidden_vertices=None)[source]¶

Return the number of connected components.

INPUT:

G – the input graph
forbidden_vertices – list (default: None); set of vertices to avoid during the search

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import connected_components_number
sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: connected_components_number(G)
2
sage: G.connected_components_number()
2
sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: connected_components_number(D)
2
sage: connected_components_number(D, forbidden_vertices=[1, 3])
3

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import connected_components_number
>>> G = Graph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> connected_components_number(G)
2
>>> G.connected_components_number()
2
>>> D = DiGraph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> connected_components_number(D)
2
>>> connected_components_number(D, forbidden_vertices=[Integer(1), Integer(3)])
3

sage.graphs.connectivity.connected_components_sizes(G, forbidden_vertices=None)[source]¶

Return the sizes of the connected components as a list.

The list is sorted from largest to lower values.

INPUT:

G – the input graph
forbidden_vertices – list (default: None); set of vertices to avoid during the search

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import connected_components_sizes
sage: for x in graphs(3):
....:     print(connected_components_sizes(x))
[1, 1, 1]
[2, 1]
[3]
[3]
sage: for x in graphs(3):
....:     print(x.connected_components_sizes())
[1, 1, 1]
[2, 1]
[3]
[3]
sage: G = graphs.PathGraph(5)
sage: G.connected_components_sizes()
[5]
sage: G.connected_components_sizes(forbidden_vertices=[1])
[3, 1]
sage: G.connected_components_sizes(forbidden_vertices=[1, 3])
[1, 1, 1]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import connected_components_sizes
>>> for x in graphs(Integer(3)):
...     print(connected_components_sizes(x))
[1, 1, 1]
[2, 1]
[3]
[3]
>>> for x in graphs(Integer(3)):
...     print(x.connected_components_sizes())
[1, 1, 1]
[2, 1]
[3]
[3]
>>> G = graphs.PathGraph(Integer(5))
>>> G.connected_components_sizes()
[5]
>>> G.connected_components_sizes(forbidden_vertices=[Integer(1)])
[3, 1]
>>> G.connected_components_sizes(forbidden_vertices=[Integer(1), Integer(3)])
[1, 1, 1]

sage.graphs.connectivity.connected_components_subgraphs(G, forbidden_vertices=None)[source]¶

Return a list of connected components as graph objects.

INPUT:

G – the input graph
forbidden_vertices – list (default: None); set of vertices to avoid during the search

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import connected_components_subgraphs
sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: L = connected_components_subgraphs(G)
sage: graphs_list.show_graphs(L)                                                # needs sage.plot
sage: L = connected_components_subgraphs(G, forbidden_vertices=[1, 3])
sage: graphs_list.show_graphs(L)                                                # needs sage.plot
sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]})
sage: L = connected_components_subgraphs(D)
sage: graphs_list.show_graphs(L)                                                # needs sage.plot
sage: L = D.connected_components_subgraphs()
sage: graphs_list.show_graphs(L)                                                # needs sage.plot

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import connected_components_subgraphs
>>> G = Graph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> L = connected_components_subgraphs(G)
>>> graphs_list.show_graphs(L)                                                # needs sage.plot
>>> L = connected_components_subgraphs(G, forbidden_vertices=[Integer(1), Integer(3)])
>>> graphs_list.show_graphs(L)                                                # needs sage.plot
>>> D = DiGraph({Integer(0): [Integer(1), Integer(3)], Integer(1): [Integer(2)], Integer(2): [Integer(3)], Integer(4): [Integer(5), Integer(6)], Integer(5): [Integer(6)]})
>>> L = connected_components_subgraphs(D)
>>> graphs_list.show_graphs(L)                                                # needs sage.plot
>>> L = D.connected_components_subgraphs()
>>> graphs_list.show_graphs(L)                                                # needs sage.plot

sage.graphs.connectivity.edge_connectivity(G, value_only=True, implementation=None, use_edge_labels=False, vertices=False, solver=None, verbose=0, integrality_tolerance=0.001)[source]¶

Return the edge connectivity of the graph.

For more information, see the Wikipedia article Connectivity_(graph_theory).

Note

When the graph is a directed graph, this method actually computes the strong connectivity, (i.e. a directed graph is strongly \(k\)-connected if there are \(k\) disjoint paths between any two vertices \(u, v\)). If you do not want to consider strong connectivity, the best is probably to convert your DiGraph object to a Graph object, and compute the connectivity of this other graph.

INPUT:

G – the input Sage (Di)Graph
value_only – boolean (default: True)
- When set to True (default), only the value is returned.
- When set to False, both the value and a minimum vertex cut are returned.
implementation – string (default: None); selects an implementation:
- None – default; selects the best implementation available
- 'boost' – use the Boost graph library (which is much more efficient). It is not available when edge_labels=True, and it is unreliable for directed graphs (see Issue #18753).
- 'Sage' – use Sage’s implementation based on integer linear programming
use_edge_labels – boolean (default: False)
- When set to True, computes a weighted minimum cut where each edge has a weight defined by its label. (If an edge has no label, \(1\) is assumed.). Implies boost = False.
- When set to False, each edge has weight \(1\).
vertices – boolean (default: False)
- When set to True, also returns the two sets of vertices that are disconnected by the cut. Implies value_only=False.
solver – string (default: None); specifies a Mixed Integer Linear Programming (MILP) solver to be used. If set to None, the default one is used. For more information on MILP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
verbose – integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.
integrality_tolerance – float; parameter for use with MILP solvers over an inexact base ring; see MixedIntegerLinearProgram.get_values().

EXAMPLES:

A basic application on the PappusGraph:

Sage

sage: from sage.graphs.connectivity import edge_connectivity
sage: g = graphs.PappusGraph()
sage: edge_connectivity(g)
3
sage: g.edge_connectivity()
3

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import edge_connectivity
>>> g = graphs.PappusGraph()
>>> edge_connectivity(g)
3
>>> g.edge_connectivity()
3

The edge connectivity of a complete graph is its minimum degree, and one of the two parts of the bipartition is reduced to only one vertex. The graph of the cut edges is isomorphic to a Star graph:

Sage

sage: g = graphs.CompleteGraph(5)
sage: [ value, edges, [ setA, setB ]] = edge_connectivity(g,vertices=True)
sage: value
4
sage: len(setA) == 1 or len(setB) == 1
True
sage: cut = Graph()
sage: cut.add_edges(edges)
sage: cut.is_isomorphic(graphs.StarGraph(4))
True

Python

>>> from sage.all import *
>>> g = graphs.CompleteGraph(Integer(5))
>>> [ value, edges, [ setA, setB ]] = edge_connectivity(g,vertices=True)
>>> value
4
>>> len(setA) == Integer(1) or len(setB) == Integer(1)
True
>>> cut = Graph()
>>> cut.add_edges(edges)
>>> cut.is_isomorphic(graphs.StarGraph(Integer(4)))
True

Even if obviously in any graph we know that the edge connectivity is less than the minimum degree of the graph:

Sage

sage: g = graphs.RandomGNP(10,.3)
sage: min(g.degree()) >= edge_connectivity(g)
True

Python

>>> from sage.all import *
>>> g = graphs.RandomGNP(Integer(10),RealNumber('.3'))
>>> min(g.degree()) >= edge_connectivity(g)
True

If we build a tree then assign to its edges a random value, the minimum cut will be the edge with minimum value:

Sage

sage: tree = graphs.RandomTree(10)
sage: for u,v in tree.edge_iterator(labels=None):
....:      tree.set_edge_label(u, v, random())
sage: minimum = min(tree.edge_labels())
sage: [_, [(_, _, l)]] = edge_connectivity(tree, value_only=False,              # needs sage.numerical.mip
....:                                      use_edge_labels=True)
sage: l == minimum                                                              # needs sage.numerical.mip
True

Python

>>> from sage.all import *
>>> tree = graphs.RandomTree(Integer(10))
>>> for u,v in tree.edge_iterator(labels=None):
...      tree.set_edge_label(u, v, random())
>>> minimum = min(tree.edge_labels())
>>> [_, [(_, _, l)]] = edge_connectivity(tree, value_only=False,              # needs sage.numerical.mip
...                                      use_edge_labels=True)
>>> l == minimum                                                              # needs sage.numerical.mip
True

When value_only=True and implementation="sage", this function is optimized for small connectivity values and does not need to build a linear program.

It is the case for graphs which are not connected

Sage

sage: g = 2 * graphs.PetersenGraph()
sage: edge_connectivity(g, implementation='sage')
0.0

Python

>>> from sage.all import *
>>> g = Integer(2) * graphs.PetersenGraph()
>>> edge_connectivity(g, implementation='sage')
0.0

For directed graphs, the strong connectivity is tested through the dedicated function:

Sage

sage: g = digraphs.ButterflyGraph(3)
sage: edge_connectivity(g, implementation='sage')
0.0

Python

>>> from sage.all import *
>>> g = digraphs.ButterflyGraph(Integer(3))
>>> edge_connectivity(g, implementation='sage')
0.0

We check that the result with Boost is the same as the result without Boost:

Sage

sage: g = graphs.RandomGNP(15, .3)
sage: (edge_connectivity(g, implementation='boost')                             # needs sage.numerical.mip
....:    == edge_connectivity(g, implementation='sage'))
True

Python

>>> from sage.all import *
>>> g = graphs.RandomGNP(Integer(15), RealNumber('.3'))
>>> (edge_connectivity(g, implementation='boost')                             # needs sage.numerical.mip
...    == edge_connectivity(g, implementation='sage'))
True

Boost interface also works with directed graphs:

Sage

sage: edge_connectivity(digraphs.Circuit(10), implementation='boost',
....:                   vertices=True)
[1, [(0, 1)], [{0}, {1, 2, 3, 4, 5, 6, 7, 8, 9}]]

Python

>>> from sage.all import *
>>> edge_connectivity(digraphs.Circuit(Integer(10)), implementation='boost',
...                   vertices=True)
[1, [(0, 1)], [{0}, {1, 2, 3, 4, 5, 6, 7, 8, 9}]]

However, the Boost algorithm is not reliable if the input is directed (see Issue #18753):

Sage

sage: g = digraphs.Path(3)
sage: edge_connectivity(g)
0.0
sage: edge_connectivity(g, implementation='boost')
1
sage: g.add_edge(1, 0)
sage: edge_connectivity(g)
0.0
sage: edge_connectivity(g, implementation='boost')
0

Python

>>> from sage.all import *
>>> g = digraphs.Path(Integer(3))
>>> edge_connectivity(g)
0.0
>>> edge_connectivity(g, implementation='boost')
1
>>> g.add_edge(Integer(1), Integer(0))
>>> edge_connectivity(g)
0.0
>>> edge_connectivity(g, implementation='boost')
0

sage.graphs.connectivity.is_connected(G, forbidden_vertices=None)[source]¶

Check whether the (di)graph is connected.

Note that in a graph, path connected is equivalent to connected.

INPUT:

G – the input graph
forbidden_vertices – list (default: None); set of vertices to avoid during the search

See also

is_biconnected()

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import is_connected
sage: G = Graph({0: [1, 2], 1: [2], 3: [4, 5], 4: [5]})
sage: is_connected(G)
False
sage: G.is_connected()
False
sage: G.add_edge(0,3)
sage: is_connected(G)
True
sage: is_connected(G, forbidden_vertices=[3])
False
sage: is_connected(G, forbidden_vertices=[1])
True
sage: D = DiGraph({0: [1, 2], 1: [2], 3: [4, 5], 4: [5]})
sage: is_connected(D)
False
sage: D.add_edge(0, 3)
sage: is_connected(D)
True
sage: D = DiGraph({1: [0], 2: [0]})
sage: is_connected(D)
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_connected
>>> G = Graph({Integer(0): [Integer(1), Integer(2)], Integer(1): [Integer(2)], Integer(3): [Integer(4), Integer(5)], Integer(4): [Integer(5)]})
>>> is_connected(G)
False
>>> G.is_connected()
False
>>> G.add_edge(Integer(0),Integer(3))
>>> is_connected(G)
True
>>> is_connected(G, forbidden_vertices=[Integer(3)])
False
>>> is_connected(G, forbidden_vertices=[Integer(1)])
True
>>> D = DiGraph({Integer(0): [Integer(1), Integer(2)], Integer(1): [Integer(2)], Integer(3): [Integer(4), Integer(5)], Integer(4): [Integer(5)]})
>>> is_connected(D)
False
>>> D.add_edge(Integer(0), Integer(3))
>>> is_connected(D)
True
>>> D = DiGraph({Integer(1): [Integer(0)], Integer(2): [Integer(0)]})
>>> is_connected(D)
True

sage.graphs.connectivity.is_cut_edge(G, u, v=None, label=None)[source]¶

Check whether the edge (u, v) is a cut-edge or a bridge of graph G.

A cut edge (or bridge) is an edge that when removed increases the number of connected components. This function works with simple graphs as well as graphs with loops and multiedges. In a digraph, a cut edge is an edge that when removed increases the number of (weakly) connected components.

INPUT: The following forms are accepted

is_cut_edge(G, 1, 2 )
is_cut_edge(G, (1, 2) )
is_cut_edge(G, 1, 2, ‘label’ )
is_cut_edge(G, (1, 2, ‘label’) )

OUTPUT:

Returns True if (u,v) is a cut edge, False otherwise

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import is_cut_edge
sage: G = graphs.CompleteGraph(4)
sage: is_cut_edge(G,0,2)
False
sage: G.is_cut_edge(0,2)
False

sage: G = graphs.CompleteGraph(4)
sage: G.add_edge((0,5,'silly'))
sage: is_cut_edge(G,(0,5,'silly'))
True

sage: G = Graph([[0,1],[0,2],[3,4],[4,5],[3,5]])
sage: is_cut_edge(G,(0,1))
True

sage: G = Graph([[0,1],[0,2],[1,1]], loops = True)
sage: is_cut_edge(G,(1,1))
False

sage: G = digraphs.Circuit(5)
sage: is_cut_edge(G,(0,1))
False

sage: G = graphs.CompleteGraph(6)
sage: is_cut_edge(G,(0,7))
Traceback (most recent call last):
...
ValueError: edge not in graph

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_cut_edge
>>> G = graphs.CompleteGraph(Integer(4))
>>> is_cut_edge(G,Integer(0),Integer(2))
False
>>> G.is_cut_edge(Integer(0),Integer(2))
False

>>> G = graphs.CompleteGraph(Integer(4))
>>> G.add_edge((Integer(0),Integer(5),'silly'))
>>> is_cut_edge(G,(Integer(0),Integer(5),'silly'))
True

>>> G = Graph([[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(3),Integer(4)],[Integer(4),Integer(5)],[Integer(3),Integer(5)]])
>>> is_cut_edge(G,(Integer(0),Integer(1)))
True

>>> G = Graph([[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(1),Integer(1)]], loops = True)
>>> is_cut_edge(G,(Integer(1),Integer(1)))
False

>>> G = digraphs.Circuit(Integer(5))
>>> is_cut_edge(G,(Integer(0),Integer(1)))
False

>>> G = graphs.CompleteGraph(Integer(6))
>>> is_cut_edge(G,(Integer(0),Integer(7)))
Traceback (most recent call last):
...
ValueError: edge not in graph

sage.graphs.connectivity.is_cut_vertex(G, u, weak=False)[source]¶

Check whether the input vertex is a cut-vertex.

A vertex is a cut-vertex if its removal from the (di)graph increases the number of (strongly) connected components. Isolated vertices or leaves are not cut-vertices. This function works with simple graphs as well as graphs with loops and multiple edges.

INPUT:

G – a Sage (Di)Graph
u – a vertex
weak – boolean (default: False); whether the connectivity of directed graphs is to be taken in the weak sense, that is ignoring edges orientations

OUTPUT:

Return True if u is a cut-vertex, and False otherwise.

EXAMPLES:

Giving a LollipopGraph(4,2), that is a complete graph with 4 vertices with a pending edge:

Sage

sage: from sage.graphs.connectivity import is_cut_vertex
sage: G = graphs.LollipopGraph(4, 2)
sage: is_cut_vertex(G, 0)
False
sage: is_cut_vertex(G, 3)
True
sage: G.is_cut_vertex(3)
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_cut_vertex
>>> G = graphs.LollipopGraph(Integer(4), Integer(2))
>>> is_cut_vertex(G, Integer(0))
False
>>> is_cut_vertex(G, Integer(3))
True
>>> G.is_cut_vertex(Integer(3))
True

Comparing the weak and strong connectivity of a digraph:

Sage

sage: D = digraphs.Circuit(6)
sage: D.is_strongly_connected()
True
sage: is_cut_vertex(D, 2)
True
sage: is_cut_vertex(D, 2, weak=True)
False

Python

>>> from sage.all import *
>>> D = digraphs.Circuit(Integer(6))
>>> D.is_strongly_connected()
True
>>> is_cut_vertex(D, Integer(2))
True
>>> is_cut_vertex(D, Integer(2), weak=True)
False

Giving a vertex that is not in the graph:

Sage

sage: G = graphs.CompleteGraph(4)
sage: is_cut_vertex(G, 7)
Traceback (most recent call last):
...
ValueError: vertex (7) is not a vertex of the graph

Python

>>> from sage.all import *
>>> G = graphs.CompleteGraph(Integer(4))
>>> is_cut_vertex(G, Integer(7))
Traceback (most recent call last):
...
ValueError: vertex (7) is not a vertex of the graph

sage.graphs.connectivity.is_edge_cut(G, edges)[source]¶

Check whether edges form an edge cut.

A set of edges is an edge cut of a graph if its removal increases the number of connected components. In a digraph, we consider the number of (weakly) connected components.

This method is not working for (di)graphs with multiple edges. Furthermore, edge labels are ignored.

INPUT:

G – a (di)graph
edges – a set of edges

EXAMPLES:

A cycle graph of order 4:

Sage

sage: from sage.graphs.connectivity import is_edge_cut
sage: G = graphs.CycleGraph(4)
sage: is_edge_cut(G, [(1, 2)])
False
sage: is_edge_cut(G, [(1, 2), (2, 3)])
True
sage: is_edge_cut(G, [(1, 2), (3, 0)])
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_edge_cut
>>> G = graphs.CycleGraph(Integer(4))
>>> is_edge_cut(G, [(Integer(1), Integer(2))])
False
>>> is_edge_cut(G, [(Integer(1), Integer(2)), (Integer(2), Integer(3))])
True
>>> is_edge_cut(G, [(Integer(1), Integer(2)), (Integer(3), Integer(0))])
True

A pending edge is a cut-edge:

Sage

sage: G.add_edge((0, 5, 'silly'))
sage: is_edge_cut(G, [(0, 5, 'silly')])
True

Python

>>> from sage.all import *
>>> G.add_edge((Integer(0), Integer(5), 'silly'))
>>> is_edge_cut(G, [(Integer(0), Integer(5), 'silly')])
True

Edge labels are ignored, even if specified:

Sage

sage: G.add_edge((2, 5, 'xyz'))
sage: is_edge_cut(G, [(0, 5), (2, 5)])
True
sage: is_edge_cut(G, [(0, 5), (2, 5, 'xyz')])
True
sage: is_edge_cut(G, [(0, 5, 'silly'), (2, 5)])
True
sage: is_edge_cut(G, [(0, 5, 'aa'), (2, 5, 'bb')])
True

Python

>>> from sage.all import *
>>> G.add_edge((Integer(2), Integer(5), 'xyz'))
>>> is_edge_cut(G, [(Integer(0), Integer(5)), (Integer(2), Integer(5))])
True
>>> is_edge_cut(G, [(Integer(0), Integer(5)), (Integer(2), Integer(5), 'xyz')])
True
>>> is_edge_cut(G, [(Integer(0), Integer(5), 'silly'), (Integer(2), Integer(5))])
True
>>> is_edge_cut(G, [(Integer(0), Integer(5), 'aa'), (Integer(2), Integer(5), 'bb')])
True

The graph can have loops:

Sage

sage: G.allow_loops(True)
sage: G.add_edge(0, 0)
sage: is_edge_cut(G, [(0, 5), (2, 5)])
True
sage: is_edge_cut(G, [(0, 0), (0, 5), (2, 5)])
True

Python

>>> from sage.all import *
>>> G.allow_loops(True)
>>> G.add_edge(Integer(0), Integer(0))
>>> is_edge_cut(G, [(Integer(0), Integer(5)), (Integer(2), Integer(5))])
True
>>> is_edge_cut(G, [(Integer(0), Integer(0)), (Integer(0), Integer(5)), (Integer(2), Integer(5))])
True

Multiple edges are not allowed:

Sage

sage: G.allow_multiple_edges(True)
sage: is_edge_cut(G, [(0, 5), (2, 5)])
Traceback (most recent call last):
...
ValueError: This method is not known to work on graphs with
 multiedges. Perhaps this method can be updated to handle them, but in
 the meantime if you want to use it please disallow multiedges using
 allow_multiple_edges().

Python

>>> from sage.all import *
>>> G.allow_multiple_edges(True)
>>> is_edge_cut(G, [(Integer(0), Integer(5)), (Integer(2), Integer(5))])
Traceback (most recent call last):
...
ValueError: This method is not known to work on graphs with
 multiedges. Perhaps this method can be updated to handle them, but in
 the meantime if you want to use it please disallow multiedges using
 allow_multiple_edges().

An error is raised if an element of edges is not an edge of \(G\):

Sage

sage: G = graphs.CycleGraph(4)
sage: is_edge_cut(G, [(0, 2)])
Traceback (most recent call last):
...
ValueError: edge (0, 2) is not an edge of the graph

Python

>>> from sage.all import *
>>> G = graphs.CycleGraph(Integer(4))
>>> is_edge_cut(G, [(Integer(0), Integer(2))])
Traceback (most recent call last):
...
ValueError: edge (0, 2) is not an edge of the graph

For digraphs, this method considers the number of (weakly) connected components:

Sage

sage: G = digraphs.Circuit(4)
sage: is_edge_cut(G, [(0, 1)])
False
sage: G = digraphs.Circuit(4)
sage: is_edge_cut(G, [(0, 1), (1, 2)])
True

Python

>>> from sage.all import *
>>> G = digraphs.Circuit(Integer(4))
>>> is_edge_cut(G, [(Integer(0), Integer(1))])
False
>>> G = digraphs.Circuit(Integer(4))
>>> is_edge_cut(G, [(Integer(0), Integer(1)), (Integer(1), Integer(2))])
True

For disconnected (di)graphs, the method checks if the number of (weakly) connected components increases:

Sage

sage: G = graphs.CycleGraph(4) * 2
sage: is_edge_cut(G, [(1, 2), (2, 3)])
True
sage: G = digraphs.Circuit(4) * 2
sage: is_edge_cut(G, [(0, 1), (1, 2)])
True

Python

>>> from sage.all import *
>>> G = graphs.CycleGraph(Integer(4)) * Integer(2)
>>> is_edge_cut(G, [(Integer(1), Integer(2)), (Integer(2), Integer(3))])
True
>>> G = digraphs.Circuit(Integer(4)) * Integer(2)
>>> is_edge_cut(G, [(Integer(0), Integer(1)), (Integer(1), Integer(2))])
True

sage.graphs.connectivity.is_strongly_connected(G)[source]¶

Check whether the current DiGraph is strongly connected.

EXAMPLES:

The circuit is obviously strongly connected:

Sage

sage: from sage.graphs.connectivity import is_strongly_connected
sage: g = digraphs.Circuit(5)
sage: is_strongly_connected(g)
True
sage: g.is_strongly_connected()
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_strongly_connected
>>> g = digraphs.Circuit(Integer(5))
>>> is_strongly_connected(g)
True
>>> g.is_strongly_connected()
True

But a transitive triangle is not:

Sage

sage: g = DiGraph({0: [1, 2], 1: [2]})
sage: is_strongly_connected(g)
False

Python

>>> from sage.all import *
>>> g = DiGraph({Integer(0): [Integer(1), Integer(2)], Integer(1): [Integer(2)]})
>>> is_strongly_connected(g)
False

sage.graphs.connectivity.is_triconnected(G)[source]¶

Check whether the graph is triconnected.

A triconnected graph is a connected graph on 3 or more vertices that is not broken into disconnected pieces by deleting any pair of vertices.

EXAMPLES:

The Petersen graph is triconnected:

Sage

sage: G = graphs.PetersenGraph()
sage: G.is_triconnected()
True

Python

>>> from sage.all import *
>>> G = graphs.PetersenGraph()
>>> G.is_triconnected()
True

But a 2D grid is not:

Sage

sage: G = graphs.Grid2dGraph(3, 3)
sage: G.is_triconnected()
False

Python

>>> from sage.all import *
>>> G = graphs.Grid2dGraph(Integer(3), Integer(3))
>>> G.is_triconnected()
False

By convention, a cycle of order 3 is triconnected:

Sage

sage: G = graphs.CycleGraph(3)
sage: G.is_triconnected()
True

Python

>>> from sage.all import *
>>> G = graphs.CycleGraph(Integer(3))
>>> G.is_triconnected()
True

But cycles of order 4 and more are not:

Sage

sage: [graphs.CycleGraph(i).is_triconnected() for i in range(4, 8)]
[False, False, False, False]

Python

>>> from sage.all import *
>>> [graphs.CycleGraph(i).is_triconnected() for i in range(Integer(4), Integer(8))]
[False, False, False, False]

Comparing different methods on random graphs that are not always triconnected:

Sage

sage: G = graphs.RandomBarabasiAlbert(50, 3)                                    # needs networkx
sage: G.is_triconnected() == G.vertex_connectivity(k=3)                         # needs networkx
True

Python

>>> from sage.all import *
>>> G = graphs.RandomBarabasiAlbert(Integer(50), Integer(3))                                    # needs networkx
>>> G.is_triconnected() == G.vertex_connectivity(k=Integer(3))                         # needs networkx
True

See also

sage.graphs.connectivity.is_vertex_cut(G, cut, weak=False)[source]¶

Check whether the input vertices form a vertex cut.

A set of vertices is a vertex cut if its removal from the (di)graph increases the number of (strongly) connected components. This function works with simple graphs as well as graphs with loops and multiple edges.

INPUT:

G – a Sage (Di)Graph
cut – a set of vertices
weak – boolean (default: False); whether the connectivity of directed graphs is to be taken in the weak sense, that is ignoring edges orientations

EXAMPLES:

Giving a cycle graph of order 4:

Sage

sage: from sage.graphs.connectivity import is_vertex_cut
sage: G = graphs.CycleGraph(4)
sage: is_vertex_cut(G, [0, 1])
False
sage: is_vertex_cut(G, [0, 2])
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_vertex_cut
>>> G = graphs.CycleGraph(Integer(4))
>>> is_vertex_cut(G, [Integer(0), Integer(1)])
False
>>> is_vertex_cut(G, [Integer(0), Integer(2)])
True

Giving a disconnected graph:

Sage

sage: from sage.graphs.connectivity import is_vertex_cut
sage: G = graphs.CycleGraph(4) * 2
sage: G.connected_components()
[[0, 1, 2, 3], [4, 5, 6, 7]]
sage: is_vertex_cut(G, [0, 2])
True
sage: is_vertex_cut(G, [4, 6])
True
sage: is_vertex_cut(G, [0, 6])
False
sage: is_vertex_cut(G, [0, 4, 6])
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import is_vertex_cut
>>> G = graphs.CycleGraph(Integer(4)) * Integer(2)
>>> G.connected_components()
[[0, 1, 2, 3], [4, 5, 6, 7]]
>>> is_vertex_cut(G, [Integer(0), Integer(2)])
True
>>> is_vertex_cut(G, [Integer(4), Integer(6)])
True
>>> is_vertex_cut(G, [Integer(0), Integer(6)])
False
>>> is_vertex_cut(G, [Integer(0), Integer(4), Integer(6)])
True

Comparing the weak and strong connectivity of a digraph:

Sage

sage: D = digraphs.Circuit(6)
sage: D.is_strongly_connected()
True
sage: is_vertex_cut(D, [2])
True
sage: is_vertex_cut(D, [2], weak=True)
False

Python

>>> from sage.all import *
>>> D = digraphs.Circuit(Integer(6))
>>> D.is_strongly_connected()
True
>>> is_vertex_cut(D, [Integer(2)])
True
>>> is_vertex_cut(D, [Integer(2)], weak=True)
False

Giving a vertex that is not in the graph:

Sage

sage: G = graphs.CompleteGraph(4)
sage: is_vertex_cut(G, [7])
Traceback (most recent call last):
...
ValueError: vertex (7) is not a vertex of the graph

Python

>>> from sage.all import *
>>> G = graphs.CompleteGraph(Integer(4))
>>> is_vertex_cut(G, [Integer(7)])
Traceback (most recent call last):
...
ValueError: vertex (7) is not a vertex of the graph

sage.graphs.connectivity.minimal_separators(G, forbidden_vertices=None)[source]¶

Return an iterator over the minimal separators of G.

A separator in a graph is a set of vertices whose removal increases the number of connected components. In other words, a separator is a vertex cut. This method implements the algorithm proposed in [BBC2000]. It computes the set \(S\) of minimal separators of a graph in \(O(n^3)\) time per separator, and so overall in \(O(n^3 |S|)\) time.

Warning

Note that all separators are recorded during the execution of the algorithm and so the memory consumption of this method might be huge.

INPUT:

G – an undirected graph
forbidden_vertices – list (default: None); set of vertices to avoid during the search

EXAMPLES:

Sage

sage: P = graphs.PathGraph(5)
sage: sorted(sorted(sep) for sep in P.minimal_separators())
[[1], [2], [3]]
sage: C = graphs.CycleGraph(6)
sage: sorted(sorted(sep) for sep in C.minimal_separators())
[[0, 2], [0, 3], [0, 4], [1, 3], [1, 4], [1, 5], [2, 4], [2, 5], [3, 5]]
sage: sorted(sorted(sep) for sep in C.minimal_separators(forbidden_vertices=[0]))
[[2], [3], [4]]
sage: sorted(sorted(sep) for sep in (P + C).minimal_separators())
[[1], [2], [3], [5, 7], [5, 8], [5, 9], [6, 8],
 [6, 9], [6, 10], [7, 9], [7, 10], [8, 10]]
sage: sorted(sorted(sep) for sep in (P + C).minimal_separators(forbidden_vertices=[10]))
[[1], [2], [3], [6], [7], [8]]

sage: G = graphs.RandomGNP(10, .3)
sage: all(G.is_vertex_cut(sep) for sep in G.minimal_separators())
True

Python

>>> from sage.all import *
>>> P = graphs.PathGraph(Integer(5))
>>> sorted(sorted(sep) for sep in P.minimal_separators())
[[1], [2], [3]]
>>> C = graphs.CycleGraph(Integer(6))
>>> sorted(sorted(sep) for sep in C.minimal_separators())
[[0, 2], [0, 3], [0, 4], [1, 3], [1, 4], [1, 5], [2, 4], [2, 5], [3, 5]]
>>> sorted(sorted(sep) for sep in C.minimal_separators(forbidden_vertices=[Integer(0)]))
[[2], [3], [4]]
>>> sorted(sorted(sep) for sep in (P + C).minimal_separators())
[[1], [2], [3], [5, 7], [5, 8], [5, 9], [6, 8],
 [6, 9], [6, 10], [7, 9], [7, 10], [8, 10]]
>>> sorted(sorted(sep) for sep in (P + C).minimal_separators(forbidden_vertices=[Integer(10)]))
[[1], [2], [3], [6], [7], [8]]

>>> G = graphs.RandomGNP(Integer(10), RealNumber('.3'))
>>> all(G.is_vertex_cut(sep) for sep in G.minimal_separators())
True

sage.graphs.connectivity.spqr_tree(G, algorithm='Hopcroft_Tarjan', solver=None, verbose=0, integrality_tolerance=0.001)[source]¶

Return an SPQR-tree representing the triconnected components of the graph.

'S' – the associated graph is a cycle with at least three vertices 'S' stands for series
'P' – the associated graph is a dipole graph, a multigraph with two vertices and three or more edges. 'P' stands for parallel
'Q' – the associated graph has a single real edge. This trivial case is necessary to handle the graph that has only one edge
'R' – the associated graph is a 3-connected graph that is not a cycle or dipole. 'R' stands for rigid

This method decomposes a biconnected graph into cycles, cocycles, and 3-connected blocks summed over cocycles, and arranges them as a SPQR-tree. More precisely, it splits the graph at each of its 2-vertex cuts, giving a unique decomposition into 3-connected blocks, cycles and cocycles. The cocycles are dipole graphs with one edge per real edge between the included vertices and one additional (virtual) edge per connected component resulting from deletion of the vertices in the cut. See the Wikipedia article SPQR_tree.

INPUT:

G – the input graph
algorithm – string (default: 'Hopcroft_Tarjan'); the algorithm to use among:
- 'Hopcroft_Tarjan' – default; use the algorithm proposed by Hopcroft and Tarjan in [Hopcroft1973] and later corrected by Gutwenger and Mutzel in [Gut2001]. See TriconnectivitySPQR.
- 'cleave' – using method cleave()
solver – string (default: None); specifies a Mixed Integer Linear Programming (MILP) solver to be used. If set to None, the default one is used. For more information on MILP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
verbose – integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.
integrality_tolerance – float; parameter for use with MILP solvers over an inexact base ring; see MixedIntegerLinearProgram.get_values().

OUTPUT: SPQR-tree a tree whose vertices are labeled with the block’s type and the subgraph of three-blocks in the decomposition.

EXAMPLES:

Sage

sage: from sage.graphs.connectivity import spqr_tree
sage: G = Graph(2)
sage: for i in range(3):
....:     G.add_clique([0, 1, G.add_vertex(), G.add_vertex()])
sage: Tree = spqr_tree(G)
sage: Tree.order()
4
sage: K4 = graphs.CompleteGraph(4)
sage: all(u[1].is_isomorphic(K4) for u in Tree if u[0] == 'R')
True
sage: from sage.graphs.connectivity import spqr_tree_to_graph
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G = Graph(2)
sage: for i in range(3):
....:     G.add_path([0, G.add_vertex(), G.add_vertex(), 1])
sage: Tree = spqr_tree(G)
sage: Tree.order()
4
sage: C4 = graphs.CycleGraph(4)
sage: all(u[1].is_isomorphic(C4) for u in Tree if u[0] == 'S')
True
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G.allow_multiple_edges(True)
sage: G.add_edges(G.edge_iterator())
sage: Tree = spqr_tree(G)
sage: Tree.order()
13
sage: all(u[1].is_isomorphic(C4) for u in Tree if u[0] == 'S')
True
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G = graphs.CycleGraph(6)
sage: Tree = spqr_tree(G)
sage: Tree.order()
1
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True
sage: G.add_edge(0, 3)
sage: Tree = spqr_tree(G)
sage: Tree.order()
3
sage: G.is_isomorphic(spqr_tree_to_graph(Tree))
True

sage: G = Graph('LlCG{O@?GBoMw?')
sage: T = spqr_tree(G, algorithm='Hopcroft_Tarjan')
sage: G.is_isomorphic(spqr_tree_to_graph(T))
True
sage: T2 = spqr_tree(G, algorithm='cleave')                                     # needs sage.numerical.mip
sage: G.is_isomorphic(spqr_tree_to_graph(T2))                                   # needs sage.numerical.mip
True

sage: G = Graph([(0, 1)], multiedges=True)
sage: T = spqr_tree(G, algorithm='cleave')                                      # needs sage.numerical.mip
sage: T.vertices(sort=True)                                                     # needs sage.numerical.mip
[('Q', Multi-graph on 2 vertices)]
sage: G.is_isomorphic(spqr_tree_to_graph(T))                                    # needs sage.numerical.mip
True
sage: T = spqr_tree(G, algorithm='Hopcroft_Tarjan')
sage: T.vertices(sort=True)
[('Q', Multi-graph on 2 vertices)]
sage: G.add_edge(0, 1)
sage: spqr_tree(G, algorithm='cleave').vertices(sort=True)                      # needs sage.numerical.mip
[('P', Multi-graph on 2 vertices)]

sage: from collections import Counter
sage: G = graphs.PetersenGraph()
sage: T = G.spqr_tree(algorithm='Hopcroft_Tarjan')
sage: Counter(u[0] for u in T)
Counter({'R': 1})
sage: T = G.spqr_tree(algorithm='cleave')                                       # needs sage.numerical.mip
sage: Counter(u[0] for u in T)                                                  # needs sage.numerical.mip
Counter({'R': 1})
sage: for u,v in list(G.edges(labels=False, sort=False)):
....:     G.add_path([u, G.add_vertex(), G.add_vertex(), v])
sage: T = G.spqr_tree(algorithm='Hopcroft_Tarjan')
sage: sorted(Counter(u[0] for u in T).items())
[('P', 15), ('R', 1), ('S', 15)]
sage: T = G.spqr_tree(algorithm='cleave')                                       # needs sage.numerical.mip
sage: sorted(Counter(u[0] for u in T).items())                                  # needs sage.numerical.mip
[('P', 15), ('R', 1), ('S', 15)]
sage: for u,v in list(G.edges(labels=False, sort=False)):
....:     G.add_path([u, G.add_vertex(), G.add_vertex(), v])
sage: T = G.spqr_tree(algorithm='Hopcroft_Tarjan')
sage: sorted(Counter(u[0] for u in T).items())
[('P', 60), ('R', 1), ('S', 75)]
sage: T = G.spqr_tree(algorithm='cleave')       # long time                     # needs sage.numerical.mip
sage: sorted(Counter(u[0] for u in T).items())  # long time                     # needs sage.numerical.mip
[('P', 60), ('R', 1), ('S', 75)]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import spqr_tree
>>> G = Graph(Integer(2))
>>> for i in range(Integer(3)):
...     G.add_clique([Integer(0), Integer(1), G.add_vertex(), G.add_vertex()])
>>> Tree = spqr_tree(G)
>>> Tree.order()
4
>>> K4 = graphs.CompleteGraph(Integer(4))
>>> all(u[Integer(1)].is_isomorphic(K4) for u in Tree if u[Integer(0)] == 'R')
True
>>> from sage.graphs.connectivity import spqr_tree_to_graph
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G = Graph(Integer(2))
>>> for i in range(Integer(3)):
...     G.add_path([Integer(0), G.add_vertex(), G.add_vertex(), Integer(1)])
>>> Tree = spqr_tree(G)
>>> Tree.order()
4
>>> C4 = graphs.CycleGraph(Integer(4))
>>> all(u[Integer(1)].is_isomorphic(C4) for u in Tree if u[Integer(0)] == 'S')
True
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G.allow_multiple_edges(True)
>>> G.add_edges(G.edge_iterator())
>>> Tree = spqr_tree(G)
>>> Tree.order()
13
>>> all(u[Integer(1)].is_isomorphic(C4) for u in Tree if u[Integer(0)] == 'S')
True
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G = graphs.CycleGraph(Integer(6))
>>> Tree = spqr_tree(G)
>>> Tree.order()
1
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True
>>> G.add_edge(Integer(0), Integer(3))
>>> Tree = spqr_tree(G)
>>> Tree.order()
3
>>> G.is_isomorphic(spqr_tree_to_graph(Tree))
True

>>> G = Graph('LlCG{O@?GBoMw?')
>>> T = spqr_tree(G, algorithm='Hopcroft_Tarjan')
>>> G.is_isomorphic(spqr_tree_to_graph(T))
True
>>> T2 = spqr_tree(G, algorithm='cleave')                                     # needs sage.numerical.mip
>>> G.is_isomorphic(spqr_tree_to_graph(T2))                                   # needs sage.numerical.mip
True

>>> G = Graph([(Integer(0), Integer(1))], multiedges=True)
>>> T = spqr_tree(G, algorithm='cleave')                                      # needs sage.numerical.mip
>>> T.vertices(sort=True)                                                     # needs sage.numerical.mip
[('Q', Multi-graph on 2 vertices)]
>>> G.is_isomorphic(spqr_tree_to_graph(T))                                    # needs sage.numerical.mip
True
>>> T = spqr_tree(G, algorithm='Hopcroft_Tarjan')
>>> T.vertices(sort=True)
[('Q', Multi-graph on 2 vertices)]
>>> G.add_edge(Integer(0), Integer(1))
>>> spqr_tree(G, algorithm='cleave').vertices(sort=True)                      # needs sage.numerical.mip
[('P', Multi-graph on 2 vertices)]

>>> from collections import Counter
>>> G = graphs.PetersenGraph()
>>> T = G.spqr_tree(algorithm='Hopcroft_Tarjan')
>>> Counter(u[Integer(0)] for u in T)
Counter({'R': 1})
>>> T = G.spqr_tree(algorithm='cleave')                                       # needs sage.numerical.mip
>>> Counter(u[Integer(0)] for u in T)                                                  # needs sage.numerical.mip
Counter({'R': 1})
>>> for u,v in list(G.edges(labels=False, sort=False)):
...     G.add_path([u, G.add_vertex(), G.add_vertex(), v])
>>> T = G.spqr_tree(algorithm='Hopcroft_Tarjan')
>>> sorted(Counter(u[Integer(0)] for u in T).items())
[('P', 15), ('R', 1), ('S', 15)]
>>> T = G.spqr_tree(algorithm='cleave')                                       # needs sage.numerical.mip
>>> sorted(Counter(u[Integer(0)] for u in T).items())                                  # needs sage.numerical.mip
[('P', 15), ('R', 1), ('S', 15)]
>>> for u,v in list(G.edges(labels=False, sort=False)):
...     G.add_path([u, G.add_vertex(), G.add_vertex(), v])
>>> T = G.spqr_tree(algorithm='Hopcroft_Tarjan')
>>> sorted(Counter(u[Integer(0)] for u in T).items())
[('P', 60), ('R', 1), ('S', 75)]
>>> T = G.spqr_tree(algorithm='cleave')       # long time                     # needs sage.numerical.mip
>>> sorted(Counter(u[Integer(0)] for u in T).items())  # long time                     # needs sage.numerical.mip
[('P', 60), ('R', 1), ('S', 75)]

sage.graphs.connectivity.spqr_tree_to_graph(T)[source]¶

Return the graph represented by the SPQR-tree \(T\).

The main purpose of this method is to test spqr_tree().

INPUT:

T – a SPQR tree as returned by spqr_tree()

OUTPUT: a (multi) graph

EXAMPLES:

Wikipedia article SPQR_tree reference paper example:

Sage

sage: from sage.graphs.connectivity import spqr_tree
sage: from sage.graphs.connectivity import spqr_tree_to_graph
sage: G = Graph([(1, 2), (1, 4), (1, 8), (1, 12), (3, 4), (2, 3),
....: (2, 13), (3, 13), (4, 5), (4, 7), (5, 6), (5, 8), (5, 7), (6, 7),
....: (8, 11), (8, 9), (8, 12), (9, 10), (9, 11), (9, 12), (10, 12)])
sage: T = spqr_tree(G)
sage: H = spqr_tree_to_graph(T)
sage: H.is_isomorphic(G)
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import spqr_tree
>>> from sage.graphs.connectivity import spqr_tree_to_graph
>>> G = Graph([(Integer(1), Integer(2)), (Integer(1), Integer(4)), (Integer(1), Integer(8)), (Integer(1), Integer(12)), (Integer(3), Integer(4)), (Integer(2), Integer(3)),
... (Integer(2), Integer(13)), (Integer(3), Integer(13)), (Integer(4), Integer(5)), (Integer(4), Integer(7)), (Integer(5), Integer(6)), (Integer(5), Integer(8)), (Integer(5), Integer(7)), (Integer(6), Integer(7)),
... (Integer(8), Integer(11)), (Integer(8), Integer(9)), (Integer(8), Integer(12)), (Integer(9), Integer(10)), (Integer(9), Integer(11)), (Integer(9), Integer(12)), (Integer(10), Integer(12))])
>>> T = spqr_tree(G)
>>> H = spqr_tree_to_graph(T)
>>> H.is_isomorphic(G)
True

A small multigraph

Sage

sage: G = Graph([(0, 2), (0, 2), (1, 3), (2, 3)], multiedges=True)
sage: for i in range(3):
....:     G.add_clique([0, 1, G.add_vertex(), G.add_vertex()])
sage: for i in range(3):
....:     G.add_clique([2, 3, G.add_vertex(), G.add_vertex()])
sage: T = spqr_tree(G)
sage: H = spqr_tree_to_graph(T)
sage: H.is_isomorphic(G)
True

Python

>>> from sage.all import *
>>> G = Graph([(Integer(0), Integer(2)), (Integer(0), Integer(2)), (Integer(1), Integer(3)), (Integer(2), Integer(3))], multiedges=True)
>>> for i in range(Integer(3)):
...     G.add_clique([Integer(0), Integer(1), G.add_vertex(), G.add_vertex()])
>>> for i in range(Integer(3)):
...     G.add_clique([Integer(2), Integer(3), G.add_vertex(), G.add_vertex()])
>>> T = spqr_tree(G)
>>> H = spqr_tree_to_graph(T)
>>> H.is_isomorphic(G)
True

sage.graphs.connectivity.strong_articulation_points(G)[source]¶

Return the strong articulation points of this digraph.

A vertex is a strong articulation point if its deletion increases the number of strongly connected components. This method implements the algorithm described in [ILS2012]. The time complexity is dominated by the time complexity of the immediate dominators finding algorithm.

OUTPUT: the list of strong articulation points

EXAMPLES:

Two cliques sharing a vertex:

Sage

sage: from sage.graphs.connectivity import strong_articulation_points
sage: D = digraphs.Complete(4)
sage: D.add_clique([3, 4, 5, 6])
sage: strong_articulation_points(D)
[3]
sage: D.strong_articulation_points()
[3]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import strong_articulation_points
>>> D = digraphs.Complete(Integer(4))
>>> D.add_clique([Integer(3), Integer(4), Integer(5), Integer(6)])
>>> strong_articulation_points(D)
[3]
>>> D.strong_articulation_points()
[3]

Two cliques connected by some arcs:

Sage

sage: D = digraphs.Complete(4) * 2
sage: D.add_edges([(0, 4), (7, 3)])
sage: sorted(strong_articulation_points(D))
[0, 3, 4, 7]
sage: D.add_edge(1, 5)
sage: sorted(strong_articulation_points(D))
[3, 7]
sage: D.add_edge(6, 2)
sage: strong_articulation_points(D)
[]

Python

>>> from sage.all import *
>>> D = digraphs.Complete(Integer(4)) * Integer(2)
>>> D.add_edges([(Integer(0), Integer(4)), (Integer(7), Integer(3))])
>>> sorted(strong_articulation_points(D))
[0, 3, 4, 7]
>>> D.add_edge(Integer(1), Integer(5))
>>> sorted(strong_articulation_points(D))
[3, 7]
>>> D.add_edge(Integer(6), Integer(2))
>>> strong_articulation_points(D)
[]

See also

sage.graphs.connectivity.strongly_connected_component_containing_vertex(G, v)[source]¶

Return the strongly connected component containing a given vertex.

INPUT:

G – the input DiGraph
v – a vertex

EXAMPLES:

In the symmetric digraph of a graph, the strongly connected components are the connected components:

Sage

sage: from sage.graphs.connectivity import strongly_connected_component_containing_vertex
sage: g = graphs.PetersenGraph()
sage: d = DiGraph(g)
sage: strongly_connected_component_containing_vertex(d, 0)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: d.strongly_connected_component_containing_vertex(0)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import strongly_connected_component_containing_vertex
>>> g = graphs.PetersenGraph()
>>> d = DiGraph(g)
>>> strongly_connected_component_containing_vertex(d, Integer(0))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> d.strongly_connected_component_containing_vertex(Integer(0))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Sage

sage: g = DiGraph([(0, 1), (1, 0), (1, 2), (2, 3), (3, 2)])
sage: strongly_connected_component_containing_vertex(g, 0)
[0, 1]

Python

>>> from sage.all import *
>>> g = DiGraph([(Integer(0), Integer(1)), (Integer(1), Integer(0)), (Integer(1), Integer(2)), (Integer(2), Integer(3)), (Integer(3), Integer(2))])
>>> strongly_connected_component_containing_vertex(g, Integer(0))
[0, 1]

sage.graphs.connectivity.strongly_connected_components_digraph(G, keep_labels=False)[source]¶

Return the digraph of the strongly connected components.

The digraph of the strongly connected components of a graph \(G\) has a vertex per strongly connected component included in \(G\). There is an edge from a component \(C_1\) to a component \(C_2\) if there is an edge in \(G\) from a vertex \(u_1 \in C_1\) to a vertex \(u_2 \in C_2\).

INPUT:

G – the input DiGraph
keep_labels – boolean (default: False); when keep_labels=True, the resulting digraph has an edge from a component \(C_i\) to a component \(C_j\) for each edge in \(G\) from a vertex \(u_i \in C_i\) to a vertex \(u_j \in C_j\). Hence the resulting digraph may have loops and multiple edges. However, edges in the result with same source, target, and label are not duplicated (see examples below). When keep_labels=False, the return digraph is simple, so without loops nor multiple edges, and edges are unlabelled.

EXAMPLES:

Such a digraph is always acyclic:

Sage

sage: from sage.graphs.connectivity import strongly_connected_components_digraph
sage: g = digraphs.RandomDirectedGNP(15, .1)
sage: scc_digraph = strongly_connected_components_digraph(g)
sage: scc_digraph.is_directed_acyclic()
True
sage: scc_digraph = g.strongly_connected_components_digraph()
sage: scc_digraph.is_directed_acyclic()
True

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import strongly_connected_components_digraph
>>> g = digraphs.RandomDirectedGNP(Integer(15), RealNumber('.1'))
>>> scc_digraph = strongly_connected_components_digraph(g)
>>> scc_digraph.is_directed_acyclic()
True
>>> scc_digraph = g.strongly_connected_components_digraph()
>>> scc_digraph.is_directed_acyclic()
True

The vertices of the digraph of strongly connected components are exactly the strongly connected components:

Sage

sage: g = digraphs.ButterflyGraph(2)
sage: scc_digraph = strongly_connected_components_digraph(g)
sage: g.is_directed_acyclic()
True
sage: V_scc = list(scc_digraph)
sage: all(Set(scc) in V_scc for scc in g.strongly_connected_components())
True

Python

>>> from sage.all import *
>>> g = digraphs.ButterflyGraph(Integer(2))
>>> scc_digraph = strongly_connected_components_digraph(g)
>>> g.is_directed_acyclic()
True
>>> V_scc = list(scc_digraph)
>>> all(Set(scc) in V_scc for scc in g.strongly_connected_components())
True

The following digraph has three strongly connected components, and the digraph of those is a TransitiveTournament():

Sage

sage: g = DiGraph({0: {1: "01", 2: "02", 3: "03"}, 1: {2: "12"}, 2:{1: "21", 3: "23"}})
sage: scc_digraph = strongly_connected_components_digraph(g)
sage: scc_digraph.is_isomorphic(digraphs.TransitiveTournament(3))
True

Python

>>> from sage.all import *
>>> g = DiGraph({Integer(0): {Integer(1): "01", Integer(2): "02", Integer(3): "03"}, Integer(1): {Integer(2): "12"}, Integer(2):{Integer(1): "21", Integer(3): "23"}})
>>> scc_digraph = strongly_connected_components_digraph(g)
>>> scc_digraph.is_isomorphic(digraphs.TransitiveTournament(Integer(3)))
True

By default, the labels are discarded, and the result has no loops nor multiple edges. If keep_labels is True, then the labels are kept, and the result is a multi digraph, possibly with multiple edges and loops. However, edges in the result with same source, target, and label are not duplicated (see the edges from 0 to the strongly connected component \(\{1,2\}\) below):

Sage

sage: g = DiGraph({0: {1: "0-12", 2: "0-12", 3: "0-3"}, 1: {2: "1-2", 3: "1-3"}, 2: {1: "2-1", 3: "2-3"}})
sage: g.order(), g.size()
(4, 7)
sage: scc_digraph = strongly_connected_components_digraph(g, keep_labels=True)
sage: (scc_digraph.order(), scc_digraph.size())
(3, 6)
sage: set(g.edge_labels()) == set(scc_digraph.edge_labels())
True

Python

>>> from sage.all import *
>>> g = DiGraph({Integer(0): {Integer(1): "0-12", Integer(2): "0-12", Integer(3): "0-3"}, Integer(1): {Integer(2): "1-2", Integer(3): "1-3"}, Integer(2): {Integer(1): "2-1", Integer(3): "2-3"}})
>>> g.order(), g.size()
(4, 7)
>>> scc_digraph = strongly_connected_components_digraph(g, keep_labels=True)
>>> (scc_digraph.order(), scc_digraph.size())
(3, 6)
>>> set(g.edge_labels()) == set(scc_digraph.edge_labels())
True

sage.graphs.connectivity.strongly_connected_components_subgraphs(G)[source]¶

Return the strongly connected components as a list of subgraphs.

EXAMPLES:

In the symmetric digraph of a graph, the strongly connected components are the connected components:

Sage

sage: from sage.graphs.connectivity import strongly_connected_components_subgraphs
sage: g = graphs.PetersenGraph()
sage: d = DiGraph(g)
sage: strongly_connected_components_subgraphs(d)
[Subgraph of (Petersen graph): Digraph on 10 vertices]
sage: d.strongly_connected_components_subgraphs()
[Subgraph of (Petersen graph): Digraph on 10 vertices]

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import strongly_connected_components_subgraphs
>>> g = graphs.PetersenGraph()
>>> d = DiGraph(g)
>>> strongly_connected_components_subgraphs(d)
[Subgraph of (Petersen graph): Digraph on 10 vertices]
>>> d.strongly_connected_components_subgraphs()
[Subgraph of (Petersen graph): Digraph on 10 vertices]

Sage

sage: g = DiGraph([(0, 1), (1, 0), (1, 2), (2, 3), (3, 2)])
sage: strongly_connected_components_subgraphs(g)
[Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 2 vertices]

Python

>>> from sage.all import *
>>> g = DiGraph([(Integer(0), Integer(1)), (Integer(1), Integer(0)), (Integer(1), Integer(2)), (Integer(2), Integer(3)), (Integer(3), Integer(2))])
>>> strongly_connected_components_subgraphs(g)
[Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 2 vertices]

sage.graphs.connectivity.vertex_connectivity(G, value_only=True, sets=False, k=None, solver=None, verbose=0, integrality_tolerance=0.001)[source]¶

Return the vertex connectivity of the graph.

For more information, see the Wikipedia article Connectivity_(graph_theory) and the Wikipedia article K-vertex-connected_graph.

Note

When the graph is directed, this method actually computes the strong connectivity, (i.e. a directed graph is strongly \(k\)-connected if there are \(k\) vertex disjoint paths between any two vertices \(u, v\)). If you do not want to consider strong connectivity, the best is probably to convert your DiGraph object to a Graph object, and compute the connectivity of this other graph.
By convention, a complete graph on \(n\) vertices is \(n-1\) connected. In this case, no certificate can be given as there is no pair of vertices split by a cut of order \(k-1\). For this reason, the certificates returned in this situation are empty.

INPUT:

G – the input Sage (Di)Graph
value_only – boolean (default: True)
- When set to True (default), only the value is returned.
- When set to False, both the value and a minimum vertex cut are returned.
sets – boolean (default: False); whether to also return the two
sets of vertices that are disconnected by the cut (implies value_only=False)
k – integer (default: None); when specified, check if the vertex connectivity of the (di)graph is larger or equal to \(k\). The method thus outputs a boolean only.
solver – string (default: None); specifies a Mixed Integer Linear Programming (MILP) solver to be used. If set to None, the default one is used. For more information on MILP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
verbose – integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.
integrality_tolerance – float; parameter for use with MILP solvers over an inexact base ring; see MixedIntegerLinearProgram.get_values().

EXAMPLES:

A basic application on a PappusGraph:

Sage

sage: from sage.graphs.connectivity import vertex_connectivity
sage: g = graphs.PappusGraph()
sage: vertex_connectivity(g)                                                     # needs sage.numerical.mip
3
sage: g.vertex_connectivity()                                                    # needs sage.numerical.mip
3

Python

>>> from sage.all import *
>>> from sage.graphs.connectivity import vertex_connectivity
>>> g = graphs.PappusGraph()
>>> vertex_connectivity(g)                                                     # needs sage.numerical.mip
3
>>> g.vertex_connectivity()                                                    # needs sage.numerical.mip
3

In a grid, the vertex connectivity is equal to the minimum degree, in which case one of the two sets is of cardinality \(1\):

Sage

sage: g = graphs.GridGraph([ 3,3 ])
sage: [value, cut, [ setA, setB ]] = vertex_connectivity(g, sets=True)           # needs sage.numerical.mip
sage: len(setA) == 1 or len(setB) == 1                                           # needs sage.numerical.mip
True

Python

>>> from sage.all import *
>>> g = graphs.GridGraph([ Integer(3),Integer(3) ])
>>> [value, cut, [ setA, setB ]] = vertex_connectivity(g, sets=True)           # needs sage.numerical.mip
>>> len(setA) == Integer(1) or len(setB) == Integer(1)                                           # needs sage.numerical.mip
True

A vertex cut in a tree is any internal vertex:

Sage

sage: tree = graphs.RandomTree(15)
sage: val, [cut_vertex] = vertex_connectivity(tree, value_only=False)            # needs sage.numerical.mip
sage: tree.degree(cut_vertex) > 1                                                # needs sage.numerical.mip
True

Python

>>> from sage.all import *
>>> tree = graphs.RandomTree(Integer(15))
>>> val, [cut_vertex] = vertex_connectivity(tree, value_only=False)            # needs sage.numerical.mip
>>> tree.degree(cut_vertex) > Integer(1)                                                # needs sage.numerical.mip
True

When value_only = True, this function is optimized for small connectivity values and does not need to build a linear program.

It is the case for connected graphs which are not connected:

Sage

sage: g = 2 * graphs.PetersenGraph()
sage: vertex_connectivity(g)                                                     # needs sage.numerical.mip
0

Python

>>> from sage.all import *
>>> g = Integer(2) * graphs.PetersenGraph()
>>> vertex_connectivity(g)                                                     # needs sage.numerical.mip
0

Or if they are just 1-connected:

Sage

sage: g = graphs.PathGraph(10)
sage: vertex_connectivity(g)                                                     # needs sage.numerical.mip
1

Python

>>> from sage.all import *
>>> g = graphs.PathGraph(Integer(10))
>>> vertex_connectivity(g)                                                     # needs sage.numerical.mip
1

For directed graphs, the strong connectivity is tested through the dedicated function:

Sage

sage: g = digraphs.ButterflyGraph(3)
sage: vertex_connectivity(g)                                                     # needs sage.numerical.mip
0

Python

>>> from sage.all import *
>>> g = digraphs.ButterflyGraph(Integer(3))
>>> vertex_connectivity(g)                                                     # needs sage.numerical.mip
0

A complete graph on \(10\) vertices is \(9\)-connected:

Sage

sage: g = graphs.CompleteGraph(10)
sage: vertex_connectivity(g)                                                     # needs sage.numerical.mip
9

Python

>>> from sage.all import *
>>> g = graphs.CompleteGraph(Integer(10))
>>> vertex_connectivity(g)                                                     # needs sage.numerical.mip
9

A complete digraph on \(10\) vertices is \(9\)-connected:

Sage

sage: g = DiGraph(graphs.CompleteGraph(10))
sage: vertex_connectivity(g)                                                     # needs sage.numerical.mip
9

Python

>>> from sage.all import *
>>> g = DiGraph(graphs.CompleteGraph(Integer(10)))
>>> vertex_connectivity(g)                                                     # needs sage.numerical.mip
9

When parameter k is set, we only check for the existence of a vertex cut of order at least k:

Sage

sage: g = graphs.PappusGraph()
sage: vertex_connectivity(g, k=3)                                                # needs sage.numerical.mip
True
sage: vertex_connectivity(g, k=4)                                                # needs sage.numerical.mip
False

Python

>>> from sage.all import *
>>> g = graphs.PappusGraph()
>>> vertex_connectivity(g, k=Integer(3))                                                # needs sage.numerical.mip
True
>>> vertex_connectivity(g, k=Integer(4))                                                # needs sage.numerical.mip
False