Tutte polynomial

This module implements a deletion-contraction algorithm for computing the Tutte polynomial as described in the paper [HPR2010].

tutte_polynomial()

Compute the Tutte polynomial of the input graph

Authors:

  • Mike Hansen (06-2013), Implemented the algorithm.

  • Jernej Azarija (06-2013), Tweaked the code, added documentation

Definition

Given a graph \(G\), with \(n\) vertices and \(m\) edges and \(k(G)\) connected components we define the Tutte polynomial of \(G\) as

\[\sum_H (x-1) ^{k(H) - c} (y-1)^{k(H) - |E(H)|-n}\]

where the sum ranges over all induced subgraphs \(H\) of \(G\).

Functions

class sage.graphs.tutte_polynomial.Ear(graph, end_points, interior, is_cycle)[source]

Bases: object

An ear is a sequence of vertices

Here is the definition from [HPR2010]:

An ear in a graph is a path \(v_1 - v_2 - \dots - v_n - v_{n+1}\) where \(d(v_1) > 2\), \(d(v_{n+1}) > 2\) and \(d(v_2) = d(v_3) = \dots = d(v_n) = 2\).

A cycle is viewed as a special ear where \(v_1 = v_{n+1}\) and the restriction on the degree of this vertex is lifted.

INPUT:

static find_ear(g)[source]

Finds the first ear in a graph.

EXAMPLES:

sage: G = graphs.PathGraph(4)
sage: G.add_edges([(0,4),(0,5),(3,6),(3,7)])
sage: from sage.graphs.tutte_polynomial import Ear
sage: E = Ear.find_ear(G)
sage: E.s
3
sage: E.unlabeled_edges
[(0, 1), (1, 2), (2, 3)]
sage: E.vertices
[0, 1, 2, 3]

>>> from sage.all import *
>>> G = graphs.PathGraph(Integer(4))
>>> G.add_edges([(Integer(0),Integer(4)),(Integer(0),Integer(5)),(Integer(3),Integer(6)),(Integer(3),Integer(7))])
>>> from sage.graphs.tutte_polynomial import Ear
>>> E = Ear.find_ear(G)
>>> E.s
3
>>> E.unlabeled_edges
[(0, 1), (1, 2), (2, 3)]
>>> E.vertices
[0, 1, 2, 3]
removed_from(*args, **kwds)[source]

A context manager which removes the ear from the graph \(G\).

EXAMPLES:

sage: G = graphs.PathGraph(4)
sage: G.add_edges([(0,4),(0,5),(3,6),(3,7)])
sage: len(G.edges(sort=True))
7
sage: from sage.graphs.tutte_polynomial import Ear
sage: E = Ear.find_ear(G)
sage: with E.removed_from(G) as Y:
....:     G.edges(sort=True)
[(0, 4, None), (0, 5, None), (3, 6, None), (3, 7, None)]
sage: len(G.edges(sort=True))
7

>>> from sage.all import *
>>> G = graphs.PathGraph(Integer(4))
>>> G.add_edges([(Integer(0),Integer(4)),(Integer(0),Integer(5)),(Integer(3),Integer(6)),(Integer(3),Integer(7))])
>>> len(G.edges(sort=True))
7
>>> from sage.graphs.tutte_polynomial import Ear
>>> E = Ear.find_ear(G)
>>> with E.removed_from(G) as Y:
...     G.edges(sort=True)
[(0, 4, None), (0, 5, None), (3, 6, None), (3, 7, None)]
>>> len(G.edges(sort=True))
7
property s

Return the number of distinct edges in this ear.

EXAMPLES:

sage: G = graphs.PathGraph(4)
sage: G.add_edges([(0,4),(0,5),(3,6),(3,7)])
sage: from sage.graphs.tutte_polynomial import Ear
sage: E = Ear(G,[0,3],[1,2],False)
sage: E.s
3

>>> from sage.all import *
>>> G = graphs.PathGraph(Integer(4))
>>> G.add_edges([(Integer(0),Integer(4)),(Integer(0),Integer(5)),(Integer(3),Integer(6)),(Integer(3),Integer(7))])
>>> from sage.graphs.tutte_polynomial import Ear
>>> E = Ear(G,[Integer(0),Integer(3)],[Integer(1),Integer(2)],False)
>>> E.s
3
unlabeled_edges()[source]

Return the edges in this ear.

EXAMPLES:

sage: G = graphs.PathGraph(4)
sage: G.add_edges([(0,4),(0,5),(3,6),(3,7)])
sage: from sage.graphs.tutte_polynomial import Ear
sage: E = Ear(G,[0,3],[1,2],False)
sage: E.unlabeled_edges
[(0, 1), (1, 2), (2, 3)]

>>> from sage.all import *
>>> G = graphs.PathGraph(Integer(4))
>>> G.add_edges([(Integer(0),Integer(4)),(Integer(0),Integer(5)),(Integer(3),Integer(6)),(Integer(3),Integer(7))])
>>> from sage.graphs.tutte_polynomial import Ear
>>> E = Ear(G,[Integer(0),Integer(3)],[Integer(1),Integer(2)],False)
>>> E.unlabeled_edges
[(0, 1), (1, 2), (2, 3)]
property vertices

Return the vertices of this ear.

EXAMPLES:

sage: G = graphs.PathGraph(4)
sage: G.add_edges([(0,4),(0,5),(3,6),(3,7)])
sage: from sage.graphs.tutte_polynomial import Ear
sage: E = Ear(G,[0,3],[1,2],False)
sage: E.vertices
[0, 1, 2, 3]

>>> from sage.all import *
>>> G = graphs.PathGraph(Integer(4))
>>> G.add_edges([(Integer(0),Integer(4)),(Integer(0),Integer(5)),(Integer(3),Integer(6)),(Integer(3),Integer(7))])
>>> from sage.graphs.tutte_polynomial import Ear
>>> E = Ear(G,[Integer(0),Integer(3)],[Integer(1),Integer(2)],False)
>>> E.vertices
[0, 1, 2, 3]
class sage.graphs.tutte_polynomial.EdgeSelection[source]

Bases: object

class sage.graphs.tutte_polynomial.MaximizeDegree[source]

Bases: EdgeSelection

class sage.graphs.tutte_polynomial.MinimizeDegree[source]

Bases: EdgeSelection

class sage.graphs.tutte_polynomial.MinimizeSingleDegree[source]

Bases: EdgeSelection

class sage.graphs.tutte_polynomial.VertexOrder(order)[source]

Bases: EdgeSelection

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import VertexOrder
sage: A = VertexOrder([4,6,3,2,1,7])
sage: A.order
[4, 6, 3, 2, 1, 7]
sage: A.inverse_order
{1: 4, 2: 3, 3: 2, 4: 0, 6: 1, 7: 5}

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import VertexOrder
>>> A = VertexOrder([Integer(4),Integer(6),Integer(3),Integer(2),Integer(1),Integer(7)])
>>> A.order
[4, 6, 3, 2, 1, 7]
>>> A.inverse_order
{1: 4, 2: 3, 3: 2, 4: 0, 6: 1, 7: 5}
sage.graphs.tutte_polynomial.contracted_edge(*args, **kwds)[source]

Delete the first vertex in the edge, and make all the edges that went from it go to the second vertex.

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import contracted_edge
sage: G = Graph(multiedges=True)
sage: G.add_edges([(0,1,'a'),(1,2,'b'),(0,3,'c')])
sage: G.edges(sort=True)
[(0, 1, 'a'), (0, 3, 'c'), (1, 2, 'b')]
sage: with contracted_edge(G,(0,1)) as Y:
....:     G.edges(sort=True); G.vertices(sort=True)
[(1, 2, 'b'), (1, 3, 'c')]
[1, 2, 3]
sage: G.edges(sort=True)
[(0, 1, 'a'), (0, 3, 'c'), (1, 2, 'b')]

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import contracted_edge
>>> G = Graph(multiedges=True)
>>> G.add_edges([(Integer(0),Integer(1),'a'),(Integer(1),Integer(2),'b'),(Integer(0),Integer(3),'c')])
>>> G.edges(sort=True)
[(0, 1, 'a'), (0, 3, 'c'), (1, 2, 'b')]
>>> with contracted_edge(G,(Integer(0),Integer(1))) as Y:
...     G.edges(sort=True); G.vertices(sort=True)
[(1, 2, 'b'), (1, 3, 'c')]
[1, 2, 3]
>>> G.edges(sort=True)
[(0, 1, 'a'), (0, 3, 'c'), (1, 2, 'b')]
sage.graphs.tutte_polynomial.edge_multiplicities(G)[source]

Return the dictionary of multiplicities of the edges in the graph \(G\).

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import edge_multiplicities
sage: G = Graph({1: [2,2,3], 2: [2], 3: [4,4], 4: [2,2,2]})
sage: sorted(edge_multiplicities(G).items())
[((1, 2), 2), ((1, 3), 1), ((2, 2), 1), ((2, 4), 3), ((3, 4), 2)]

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import edge_multiplicities
>>> G = Graph({Integer(1): [Integer(2),Integer(2),Integer(3)], Integer(2): [Integer(2)], Integer(3): [Integer(4),Integer(4)], Integer(4): [Integer(2),Integer(2),Integer(2)]})
>>> sorted(edge_multiplicities(G).items())
[((1, 2), 2), ((1, 3), 1), ((2, 2), 1), ((2, 4), 3), ((3, 4), 2)]
sage.graphs.tutte_polynomial.removed_edge(*args, **kwds)[source]

A context manager which removes an edge from the graph \(G\) and restores it upon exiting.

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import removed_edge
sage: G = Graph()
sage: G.add_edge(0,1)
sage: G.edges(sort=True)
[(0, 1, None)]
sage: with removed_edge(G,(0,1)) as Y:
....:     G.edges(sort=True); G.vertices(sort=True)
[]
[0, 1]
sage: G.edges(sort=True)
[(0, 1, None)]

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import removed_edge
>>> G = Graph()
>>> G.add_edge(Integer(0),Integer(1))
>>> G.edges(sort=True)
[(0, 1, None)]
>>> with removed_edge(G,(Integer(0),Integer(1))) as Y:
...     G.edges(sort=True); G.vertices(sort=True)
[]
[0, 1]
>>> G.edges(sort=True)
[(0, 1, None)]
sage.graphs.tutte_polynomial.removed_loops(*args, **kwds)[source]

A context manager which removes all the loops in the graph \(G\). It yields a list of the loops, and restores the loops upon exiting.

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import removed_loops
sage: G = Graph(multiedges=True, loops=True)
sage: G.add_edges([(0,1,'a'),(1,2,'b'),(0,0,'c')])
sage: G.edges(sort=True)
[(0, 0, 'c'), (0, 1, 'a'), (1, 2, 'b')]
sage: with removed_loops(G) as Y:
....:     G.edges(sort=True); G.vertices(sort=True); Y
[(0, 1, 'a'), (1, 2, 'b')]
[0, 1, 2]
[(0, 0, 'c')]
sage: G.edges(sort=True)
[(0, 0, 'c'), (0, 1, 'a'), (1, 2, 'b')]

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import removed_loops
>>> G = Graph(multiedges=True, loops=True)
>>> G.add_edges([(Integer(0),Integer(1),'a'),(Integer(1),Integer(2),'b'),(Integer(0),Integer(0),'c')])
>>> G.edges(sort=True)
[(0, 0, 'c'), (0, 1, 'a'), (1, 2, 'b')]
>>> with removed_loops(G) as Y:
...     G.edges(sort=True); G.vertices(sort=True); Y
[(0, 1, 'a'), (1, 2, 'b')]
[0, 1, 2]
[(0, 0, 'c')]
>>> G.edges(sort=True)
[(0, 0, 'c'), (0, 1, 'a'), (1, 2, 'b')]
sage.graphs.tutte_polynomial.removed_multiedge(*args, **kwds)[source]

A context manager which removes an edge with multiplicity from the graph \(G\) and restores it upon exiting.

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import removed_multiedge
sage: G = Graph(multiedges=True)
sage: G.add_edges([(0,1,'a'),(0,1,'b')])
sage: G.edges(sort=True)
[(0, 1, 'a'), (0, 1, 'b')]
sage: with removed_multiedge(G,(0,1)) as Y:
....:     G.edges(sort=True)
[]
sage: G.edges(sort=True)
[(0, 1, 'a'), (0, 1, 'b')]

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import removed_multiedge
>>> G = Graph(multiedges=True)
>>> G.add_edges([(Integer(0),Integer(1),'a'),(Integer(0),Integer(1),'b')])
>>> G.edges(sort=True)
[(0, 1, 'a'), (0, 1, 'b')]
>>> with removed_multiedge(G,(Integer(0),Integer(1))) as Y:
...     G.edges(sort=True)
[]
>>> G.edges(sort=True)
[(0, 1, 'a'), (0, 1, 'b')]
sage.graphs.tutte_polynomial.tutte_polynomial(G, edge_selector=None, cache=None)[source]

Return the Tutte polynomial of the graph \(G\).

INPUT:

  • edge_selector – method (optional); this argument allows the user to specify his own heuristic for selecting edges used in the deletion contraction recurrence

  • cache – (optional) dictionary to cache the Tutte polynomials generated in the recursive process. One will be created automatically if not provided.

EXAMPLES:

The Tutte polynomial of any tree of order \(n\) is \(x^{n-1}\):

sage: all(T.tutte_polynomial() == x**9 for T in graphs.trees(10))               # needs sage.symbolic
True

>>> from sage.all import *
>>> all(T.tutte_polynomial() == x**Integer(9) for T in graphs.trees(Integer(10)))               # needs sage.symbolic
True

The Tutte polynomial of the Petersen graph is:

sage: P = graphs.PetersenGraph()
sage: P.tutte_polynomial()
x^9 + 6*x^8 + 21*x^7 + 56*x^6 + 12*x^5*y + y^6 + 114*x^5 + 70*x^4*y
+ 30*x^3*y^2 + 15*x^2*y^3 + 10*x*y^4 + 9*y^5 + 170*x^4 + 170*x^3*y
+ 105*x^2*y^2 + 65*x*y^3 + 35*y^4 + 180*x^3 + 240*x^2*y + 171*x*y^2
+ 75*y^3 + 120*x^2 + 168*x*y + 84*y^2 + 36*x + 36*y

>>> from sage.all import *
>>> P = graphs.PetersenGraph()
>>> P.tutte_polynomial()
x^9 + 6*x^8 + 21*x^7 + 56*x^6 + 12*x^5*y + y^6 + 114*x^5 + 70*x^4*y
+ 30*x^3*y^2 + 15*x^2*y^3 + 10*x*y^4 + 9*y^5 + 170*x^4 + 170*x^3*y
+ 105*x^2*y^2 + 65*x*y^3 + 35*y^4 + 180*x^3 + 240*x^2*y + 171*x*y^2
+ 75*y^3 + 120*x^2 + 168*x*y + 84*y^2 + 36*x + 36*y

The Tutte polynomial of a connected graph \(G\) evaluated at (1,1) is the number of spanning trees of \(G\):

sage: G = graphs.RandomGNP(10,0.6)
sage: while not G.is_connected():
....:     G = graphs.RandomGNP(10,0.6)
sage: G.tutte_polynomial()(1,1) == G.spanning_trees_count()                     # needs sage.modules
True

>>> from sage.all import *
>>> G = graphs.RandomGNP(Integer(10),RealNumber('0.6'))
>>> while not G.is_connected():
...     G = graphs.RandomGNP(Integer(10),RealNumber('0.6'))
>>> G.tutte_polynomial()(Integer(1),Integer(1)) == G.spanning_trees_count()                     # needs sage.modules
True

Given that \(T(x,y)\) is the Tutte polynomial of a graph \(G\) with \(n\) vertices and \(c\) connected components, then \((-1)^{n-c} x^k T(1-x,0)\) is the chromatic polynomial of \(G\).

sage: G = graphs.OctahedralGraph()
sage: T = G.tutte_polynomial()
sage: R = PolynomialRing(ZZ, 'x')
sage: R((-1)^5*x*T(1-x,0)).factor()                                             # needs sage.symbolic
(x - 2) * (x - 1) * x * (x^3 - 9*x^2 + 29*x - 32)
sage: G.chromatic_polynomial().factor()                                         # needs sage.libs.flint
(x - 2) * (x - 1) * x * (x^3 - 9*x^2 + 29*x - 32)

>>> from sage.all import *
>>> G = graphs.OctahedralGraph()
>>> T = G.tutte_polynomial()
>>> R = PolynomialRing(ZZ, 'x')
>>> R((-Integer(1))**Integer(5)*x*T(Integer(1)-x,Integer(0))).factor()                                             # needs sage.symbolic
(x - 2) * (x - 1) * x * (x^3 - 9*x^2 + 29*x - 32)
>>> G.chromatic_polynomial().factor()                                         # needs sage.libs.flint
(x - 2) * (x - 1) * x * (x^3 - 9*x^2 + 29*x - 32)
sage.graphs.tutte_polynomial.underlying_graph(G)[source]

Given a graph \(G\) with multi-edges, returns a graph where all the multi-edges are replaced with a single edge.

EXAMPLES:

sage: from sage.graphs.tutte_polynomial import underlying_graph
sage: G = Graph(multiedges=True)
sage: G.add_edges([(0,1,'a'),(0,1,'b')])
sage: G.edges(sort=True)
[(0, 1, 'a'), (0, 1, 'b')]
sage: underlying_graph(G).edges(sort=True)
[(0, 1, None)]

>>> from sage.all import *
>>> from sage.graphs.tutte_polynomial import underlying_graph
>>> G = Graph(multiedges=True)
>>> G.add_edges([(Integer(0),Integer(1),'a'),(Integer(0),Integer(1),'b')])
>>> G.edges(sort=True)
[(0, 1, 'a'), (0, 1, 'b')]
>>> underlying_graph(G).edges(sort=True)
[(0, 1, None)]