# Tutte polynomial¶

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

 tutte_polynomial() Computes 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)

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)

Finds the first ear in a graph.

EXAMPLES:

sage: G = graphs.PathGraph(4)
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]

removed_from(*args, **kwds)

A context manager which removes the ear from the graph $$G$$.

EXAMPLES:

sage: G = graphs.PathGraph(4)
sage: len(G.edges())
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()
[(0, 4, None), (0, 5, None), (3, 6, None), (3, 7, None)]
sage: len(G.edges())
7

s

Returns the number of distinct edges in this ear.

EXAMPLES:

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

unlabeled_edges()

Returns the edges in this ear.

EXAMPLES:

sage: G = graphs.PathGraph(4)
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)]

vertices

Returns the vertices of this ear.

EXAMPLES:

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

class sage.graphs.tutte_polynomial.EdgeSelection

Bases: object

class sage.graphs.tutte_polynomial.MaximizeDegree
class sage.graphs.tutte_polynomial.MinimizeDegree
class sage.graphs.tutte_polynomial.MinimizeSingleDegree
class sage.graphs.tutte_polynomial.VertexOrder(order)

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}

sage.graphs.tutte_polynomial.contracted_edge(*args, **kwds)

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.edges()
[(0, 1, 'a'), (0, 3, 'c'), (1, 2, 'b')]
sage: with contracted_edge(G,(0,1)) as Y:
....:     G.edges(); G.vertices()
[(1, 2, 'b'), (1, 3, 'c')]
[1, 2, 3]
sage: G.edges()
[(0, 1, 'a'), (0, 3, 'c'), (1, 2, 'b')]

sage.graphs.tutte_polynomial.edge_multiplicities(G)

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)]

sage.graphs.tutte_polynomial.removed_edge(*args, **kwds)

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.edges()
[(0, 1, None)]
sage: with removed_edge(G,(0,1)) as Y:
....:     G.edges(); G.vertices()
[]
[0, 1]
sage: G.edges()
[(0, 1, None)]

sage.graphs.tutte_polynomial.removed_loops(*args, **kwds)

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.edges()
[(0, 0, 'c'), (0, 1, 'a'), (1, 2, 'b')]
sage: with removed_loops(G) as Y:
....:     G.edges(); G.vertices(); Y
[(0, 1, 'a'), (1, 2, 'b')]
[0, 1, 2]
[(0, 0, 'c')]
sage: G.edges()
[(0, 0, 'c'), (0, 1, 'a'), (1, 2, 'b')]

sage.graphs.tutte_polynomial.removed_multiedge(*args, **kwds)

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.edges()
[(0, 1, 'a'), (0, 1, 'b')]
sage: with removed_multiedge(G,(0,1)) as Y:
....:     G.edges()
[]
sage: G.edges()
[(0, 1, 'a'), (0, 1, 'b')]

sage.graphs.tutte_polynomial.tutte_polynomial(G, edge_selector=None, cache=None)

Return the Tutte polynomial of the graph $$G$$.

INPUT:

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


The Tutte polynomial of $$G$$ evaluated at (1,1) is the number of spanning trees of $$G$$:

sage: G = graphs.RandomGNP(10,0.6)
sage: G.tutte_polynomial()(1,1) == G.spanning_trees_count()
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()
(x - 2) * (x - 1) * x * (x^3 - 9*x^2 + 29*x - 32)
sage: G.chromatic_polynomial().factor()
(x - 2) * (x - 1) * x * (x^3 - 9*x^2 + 29*x - 32)

sage.graphs.tutte_polynomial.underlying_graph(G)

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)