# Static dense graphs#

This module gathers everything which is related to static dense graphs, i.e. :

• The vertices are integer from $$0$$ to $$n-1$$

• No labels on vertices/edges

• No multiple edges

This being said, it is technically possible to add/remove edges. The data structure does not mind at all.

It is all based on the binary matrix data structure described in data_structures/binary_matrix.pxd, which is almost a copy of the bitset data structure. The only difference is that it differentiates the rows (the vertices) instead of storing the whole data in a long bitset, and we can use that.

For an overview of graph data structures in sage, see overview.

## Index#

Cython functions

 dense_graph_init Fill a binary matrix with the information from a Sage (di)graph.

Python functions

 is_strongly_regular() Check whether the graph is strongly regular is_triangle_free() Check whether $$G$$ is triangle free triangles_count() Return the number of triangles containing $$v$$, for every $$v$$ connected_subgraph_iterator() Iterator over the induced connected subgraphs of order at most $$k$$

## Functions#

sage.graphs.base.static_dense_graph.connected_full_subgraphs(G, edges_only=False, labels=False, min_edges=None, max_edges=None)#

Return an iterator over the connected subgraphs of $$G$$ with same vertex set.

This method implements a iterator over the connected subgraphs of the input (di)graph with the same ground set of vertices. That is, it iterates over every subgraph $$H = (V_H, E_H)$$ of $$G = (V, E)$$ such that $$V_H = V$$, $$E_H \subseteq E$$ and $$H$$ is connected. Hence, this method may yield a huge number of graphs.

When the input (di)graph $$G$$ is not connected, this method returns nothing.

As for method sage.graphs.generic_graph.connected_components(), edge orientation is ignored. Hence, the directed graph with a single arc $$0 \to 1$$ is considered connected.

INPUT:

• G – a Graph or a DiGraph; loops and multiple edges are not allowed

• edges_only – boolean (default: False); whether to return (Di)Graph or list of vertices

• labels – boolean (default: False); whether to return labelled edges or not. This parameter is used only when edges_only is True.

• min_edges – integer (default: None); minimum number of edges of reported subgraphs. By default (None), this lower bound will be set to $$n - 1$$.

• max_edges – integer (default: None); maximum number of edges of reported subgraphs. By default (None), this lower bound will be set to the number of edges of the input (di)graph.

Note

Roughly, this method explores all possible subsets of neighbors of each vertex, which represents a huge number of subsets. We have thus chosen to limit the degree of the vertices of the graphs that can be considered, even if the graph has a single connected subgraph (e.g., a tree). It is therefore recommended to call this method on biconnected components, as done in connected_subgraph_iterator().

EXAMPLES:

The complete graph of order 3 has 4 connected subgraphs:

sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs
sage: G = graphs.CompleteGraph(3)
sage: len(list(connected_full_subgraphs(G)))
4


A cycle of order 5 has 6 connected subgraphs:

sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs
sage: G = graphs.CycleGraph(5)
sage: len(list(connected_full_subgraphs(G)))
6


The House graph has 18 connected subgraphs of order 5:

sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs
sage: G = graphs.HouseGraph()
sage: L = list(connected_full_subgraphs(G))
sage: len(L)
18
sage: all(g.order() == 5 for g in L)
True
sage: all(g.is_connected() for g in L)
True
sage: F = frozenset(frozenset(g.edges(sort=False, labels=False)) for g in L)
sage: len(F)
18


Specifying bounds on the number of edges:

sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs
sage: G = graphs.HouseGraph()
sage: [g.size() for g in connected_full_subgraphs(G)]
[6, 5, 5, 5, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4]
sage: [g.size() for g in connected_full_subgraphs(G, max_edges=4)]
[4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4]
sage: [g.size() for g in connected_full_subgraphs(G, min_edges=6)]

sage: [g.size() for g in connected_full_subgraphs(G, min_edges=5, max_edges=5)]
[5, 5, 5, 5, 5, 5]


sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs
sage: G = Graph([(0, 1, "01"), (0, 2, "02"), (1, 2, "12")])
sage: it = connected_full_subgraphs(G, edges_only=True)
sage: next(it)
[(0, 1), (0, 2), (1, 2)]
sage: next(it)
[(0, 1), (0, 2)]
sage: it = connected_full_subgraphs(G, edges_only=True, labels=True)
sage: next(it)
[(0, 1, '01'), (0, 2, '02'), (1, 2, '12')]
sage: next(it)
[(0, 1, '01'), (0, 2, '02')]


Subgraphs of a digraph:

sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs
sage: G = digraphs.Complete(2)
sage: list(connected_full_subgraphs(G, edges_only=True))
[[(0, 1)], [(1, 0)], [(0, 1), (1, 0)]]

sage.graphs.base.static_dense_graph.connected_subgraph_iterator(G, k=None, vertices_only=False, edges_only=False, labels=False, induced=True, exactly_k=False)#

Return an terator over the induced connected subgraphs of order at most $$k$$.

This method implements a iterator over the induced connected subgraphs of the input (di)graph. An induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset (Wikipedia article Induced_subgraph).

As for method sage.graphs.generic_graph.connected_components(), edge orientation is ignored. Hence, the directed graph with a single arc $$0 \to 1$$ is considered connected.

INPUT:

• G – a Graph or a DiGraph; loops and multiple edges are allowed

• k – (optional) integer; maximum order of the connected subgraphs to report; by default, the method iterates over all connected subgraphs (equivalent to k == n)

• vertices_only – boolean (default: False); whether to return (Di)Graph or list of vertices. This parameter is ignored when induced is True.

• edges_only – boolean (default: False); whether to return (Di)Graph or list of edges. When vertices_only is True, this parameter is ignored.

• labels – boolean (default: False); whether to return labelled edges or not. This parameter is used only when vertices_only is False and edges_only is True.

• induced – boolean (default: True); whether to return induced connected sub(di)graph only or also non-induced sub(di)graphs. This parameter can be set to False for simple (di)graphs only.

• exactly_k – boolean (default: False); True if we only return graphs of order $$k$$, False if we return graphs of order at most $$k$$.

EXAMPLES:

sage: G = DiGraph([(1, 2), (2, 3), (3, 4), (4, 2)])
sage: list(G.connected_subgraph_iterator())
[Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 3 vertices,
Subgraph of (): Digraph on 4 vertices,
Subgraph of (): Digraph on 3 vertices,
Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 3 vertices,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 1 vertex]
sage: list(G.connected_subgraph_iterator(vertices_only=True))
[, [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4],
, [2, 3], [2, 3, 4], [2, 4], , [3, 4], ]
sage: list(G.connected_subgraph_iterator(k=2))
[Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 1 vertex]
sage: list(G.connected_subgraph_iterator(k=3, vertices_only=True, exactly_k=True))
[[1, 2, 3], [1, 2, 4], [2, 3, 4]]
sage: list(G.connected_subgraph_iterator(k=2, vertices_only=True))
[, [1, 2], , [2, 3], [2, 4], , [3, 4], ]

sage: G = DiGraph([(1, 2), (2, 1)])
sage: list(G.connected_subgraph_iterator())
[Subgraph of (): Digraph on 1 vertex,
Subgraph of (): Digraph on 2 vertices,
Subgraph of (): Digraph on 1 vertex]
sage: list(G.connected_subgraph_iterator(vertices_only=True))
[, [1, 2], ]

sage: G = graphs.CompleteGraph(3)
sage: len(list(G.connected_subgraph_iterator()))
7
sage: len(list(G.connected_subgraph_iterator(vertices_only=True)))
7
sage: len(list(G.connected_subgraph_iterator(edges_only=True)))
7
sage: len(list(G.connected_subgraph_iterator(induced=False)))
10

sage: G = DiGraph([(0, 1), (1, 0), (1, 2), (2, 1)])
sage: len(list(G.connected_subgraph_iterator()))
6
sage: len(list(G.connected_subgraph_iterator(vertices_only=True)))
6
sage: len(list(G.connected_subgraph_iterator(edges_only=True)))
6
sage: len(list(G.connected_subgraph_iterator(induced=False)))
18

sage.graphs.base.static_dense_graph.is_strongly_regular(g, parameters=False)#

Check whether the graph is strongly regular.

A simple graph $$G$$ is said to be strongly regular with parameters $$(n, k, \lambda, \mu)$$ if and only if:

• $$G$$ has $$n$$ vertices

• $$G$$ is $$k$$-regular

• Any two adjacent vertices of $$G$$ have $$\lambda$$ common neighbors

• Any two non-adjacent vertices of $$G$$ have $$\mu$$ common neighbors

By convention, the complete graphs, the graphs with no edges and the empty graph are not strongly regular.

INPUT:

• parameters – boolean (default: False); whether to return the quadruple $$(n, k, \lambda, \mu)$$. If parameters = False (default), this method only returns True and False answers. If parameters = True, the True answers are replaced by quadruples $$(n, k, \lambda, \mu)$$. See definition above.

EXAMPLES:

Petersen’s graph is strongly regular:

sage: g = graphs.PetersenGraph()
sage: g.is_strongly_regular()
True
sage: g.is_strongly_regular(parameters=True)
(10, 3, 0, 1)


And Clebsch’s graph is too:

sage: g = graphs.ClebschGraph()
sage: g.is_strongly_regular()
True
sage: g.is_strongly_regular(parameters=True)
(16, 5, 0, 2)


But Chvatal’s graph is not:

sage: g = graphs.ChvatalGraph()
sage: g.is_strongly_regular()
False


Complete graphs are not strongly regular. (github issue #14297)

sage: g = graphs.CompleteGraph(5)
sage: g.is_strongly_regular()
False


Completements of complete graphs are not strongly regular:

sage: g = graphs.CompleteGraph(5).complement()
sage: g.is_strongly_regular()
False


The empty graph is not strongly regular:

sage: g = graphs.EmptyGraph()
sage: g.is_strongly_regular()
False


If the input graph has loops or multiedges an exception is raised:

sage: Graph([(1,1),(2,2)],loops=True).is_strongly_regular()
Traceback (most recent call last):
...
ValueError: This method is not known to work on graphs with
loops. Perhaps this method can be updated to handle them, but in the
meantime if you want to use it please disallow loops using
allow_loops().

sage: Graph([(1,2),(1,2)],multiedges=True).is_strongly_regular()
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().

sage.graphs.base.static_dense_graph.is_triangle_free(G, certificate=False)#

Check whether $$G$$ is triangle free.

INPUT:

• G – a Sage graph

• certificate – boolean (default: False); whether to return a triangle if one is found

EXAMPLES:

sage: from sage.graphs.base.static_dense_graph import is_triangle_free
sage: is_triangle_free(graphs.PetersenGraph())
True
sage: K4 = graphs.CompleteGraph(4)
sage: is_triangle_free(K4)
False
sage: b, certif = is_triangle_free(K4, certificate=True)
sage: K4.subgraph(certif).is_clique()
True

sage.graphs.base.static_dense_graph.triangles_count(G)#

Return the number of triangles containing $$v$$, for every $$v$$.

INPUT:

• G – a simple Sage graph

EXAMPLES:

sage: from sage.graphs.base.static_dense_graph import triangles_count
sage: triangles_count(graphs.PetersenGraph())
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0}
sage: sum(triangles_count(graphs.CompleteGraph(15)).values()) == 3 * binomial(15, 3)
True