Tutte polynomial

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This article is about the Tutte polynomial of a graph. For the Tutte polynomial of a matroid, see Matroid.
The polynomial x^4+x^3+x^2y is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane y=0, equivalent to the chromatic polynomial.

The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a polynomial in two variables which plays an important role in graph theory, a branch of mathematics and theoretical computer science. It is defined for every undirected graph and contains information about how the graph is connected.

The importance of the Tutte polynomial T_G comes from the information it contains about G. Though originally studied in algebraic graph theory as a generalisation of counting problems related to graph coloring and nowhere-zero flow, it contains several famous other specialisations from other sciences such as the Jones polynomial from knot theory and the partition functions of the Potts model from statistical physics. It is also the source of several central computational problems in theoretical computer science.

The Tutte polynomial has several equivalent definitions. It is equivalent to Whitney’s rank polynomial, Tutte’s own dichromatic polynomial and Fortuin–Kasteleyn’s random cluster model under simple transformations. It is essentially a generating function for the number of edge sets of a given size and connected components, with immediate generalisations to matroids. It is also the most general graph invariant that can be defined by a deletion–contraction recurrence. Several textbooks about graph theory and matroid theory devote entire chapters to it.[1]

Definitions[edit]

Definition. For an undirected graph G=(V,E) one may define the Tutte polynomial as

T_G(x,y)=\sum\nolimits_{A\subseteq E} (x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|},
where k(A) denotes the number of connected components of the graph (V,A). In this definition it is clear that T_G is well-defined and a polynomial in x and y.

The same definition can be given using slightly different notation by letting r(A)=|V|-k(A) denote the rank of the graph (V,A). Then the Whitney rank generating function is defined as

R_G(u,v)=\sum\nolimits_{A\subseteq E} u^{r(E)-r(A)} v^{|A|-r(A)}.

The two functions are equivalent under a simple change of variables:

T_G(x,y)=R_G(x-1,y-1).

Tutte’s dichromatic polynomial Q_G is the result of another simple transformation:

T_G(x,y)=(x-1)^{-k(G)} Q_G(x-1,y-1).

Tutte’s original definition of T_G is equivalent but less easily stated. For connected G we set

T_G(x,y)=\sum\nolimits_{i,j} t_{ij} x^iy^j,

where t_{ij} denotes the number of spanning trees of “internal activity i and external activity j.”

A third definition uses a deletion–contraction recurrence. The edge contraction G/uv of graph G is the graph obtained by merging the vertices u and v and removing the edge uv. We write G − uv for the graph where the edge uv is merely removed. Then the Tutte polynomial is defined by the recurrence relation

T_G= T_{G-e}+T_{G/e},

if e is neither a loop nor a bridge, with base case

T_G(x,y)= x^i y^j,

if G contains i bridges and j loops and no other edges. Especially, T_G=1 if G contains no edges.

The random cluster model from statistical mechanics due to Fortuin & Kasteleyn (1972) provides yet another equivalent definition.[2] The polynomial

Z_G(q,w)=\sum\nolimits_{F\subseteq E}q^{k(F)}w^{|F|}

is equivalent to T_G under the transformation[3]

T_G(x, y)=(x-1)^{-k(E)}(y-1)^{-|V|} \cdot Z_G\Big((x-1)(y-1),\; y-1\Big).

Properties[edit]

The Tutte polynomial factors into connected components: If G is the union of disjoint graphs H and H' then

T_G= T_H \cdot T_{H'}

If G is planar and G^* denotes its dual graph then

T_G(x,y)= T_{G^*} (y,x)

Especially, the chromatic polynomial of a planar graph is the flow polynomial of its dual.

Examples[edit]

Isomorphic graphs have the same Tutte polynomial, but the opposite is not true. For example, the Tutte polynomial of every tree on m edges is x^m.

Tutte polynomials are often given in tabular form by listing the coefficients t_{ij} of x^iy^j in row i and column j. For example, the Tutte polynomial of the Petersen graph,

\begin{align}
36 x &+ 120 x^2 + 180 x^3 + 170x^4+114x^5 + 56x^6 +21 x^7 + 6x^8 + x^9 \\
&+ 36y +84 y^2 + 75 y^3 +35 y^4 + 9y^5+y^6 \\
&+ 168xy + 240x^2y +170x^3y +70 x^4y + 12x^5 y \\
&+ 171xy^2+105 x^2y^2 + 30x^3y^2 \\
&+ 65xy^3 +15x^2y^3 \\
&+10xy^4,
\end{align}

is given by the following table.

0 36 84 75 35 9 1
36 168 171 65 10
120 240 105 15
180 170 30
170 70
114 12
56
21
6
1

History[edit]

W. T. Tutte’s interest in the deletion–contraction formula started in his undergraduate days at Trinity College, Cambridge, originally motivated by perfect rectangles and spanning trees. He often applied the formula in his research and “wondered if there were other interesting functions of graphs, invariant under isomorphism, with similar recursion formulae.”[4] R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte began to discover more. His original terminology for graph invariants that satisfy the delection–contraction recursion was W-function, and V-function if multiplicative over components. Tutte writes, “Playing with my W-functions I obtained a two-variable polynomial from which either the chromatic polynomial or the flow-polynomial could be obtained by setting one of the variables equal to zero, and adjusting signs.”[4] Tutte called this function the dichromate, as he saw it as a generalization of the chromatic polynomial to two variables, but it is usually referred to as the Tutte polynomial. In Tutte’s words, “This may be unfair to Hassler Whitney who knew and used analogous coefficients without bothering to affix them to two variables.” (There is “notable confusion”[5] about the terms dichromate and dichromatic polynomial, introduced by Tutte in different paper, and which differ only slightly.) The generalisation of the Tutte polynomial to matroids was first published by Crapo, though it appears already in Tutte’s thesis.[6]

Independently of the work in algebraic graph theory, Potts began studying the partition function of certain models in statistical mechanics in 1952. The work of Fortuin & Kasteleyn (1972) on the random cluster model, a generalisation of the Potts model, provided a unifying expression that showed the relation to the Tutte polynomial.[7]

Specialisations[edit]

At various points and lines of the (x,y)-plane, the Tutte polynomial evaluates to quantities that have been studied in their own right in diverse fields of mathematics and physics. Part of the appeal of the Tutte polynomial comes from the unifying framework it provides for analysing these quantities.

Chromatic polynomial[edit]

Main article: Chromatic polynomial
The chromatic polynomial drawn in the Tutte plane

At y=0, the Tutte polynomial specialises to the chromatic polynomial,

\chi_G(\lambda) = (-1)^{|V|-k(G)} \lambda^{k(G)} T_G(1-\lambda,0),

where k(G) denotes the number of connected components of G.

For integer λ the value of chromatic polynomial \chi_G(\lambda) equals the number of vertex colourings of G using a set of λ colours. It is clear that \chi_G(\lambda) does not depend on the set of colours. What is less clear is that it is the evaluation at λ of a polynomial with integer coefficients. To see this, we observe:

  1. If G has n vertices and no edges, then \chi_G(\lambda) = \lambda^n.
  2. If G contains a loop (a single edge connecting a vertex to itself), then \chi_G(\lambda) = 0.
  3. If e is an edge which is not a loop, then
\chi_G(\lambda) = \chi_{G\setminus e}(\lambda) - \chi_{G/e}(\lambda).

The three conditions above enable us to calculate \chi_G(\lambda), by applying a sequence of edge deletions and contractions; but they give no guarantee that a different sequence of deletions and contractions will lead to the same value. The guarantee comes from the fact that \chi_G(\lambda) counts something, independently of the recurrence. In particular,

T_G(2,0) = (-1)^{|V|} \chi_G(-1)

gives the number of acyclic orientations.

Jones polynomial[edit]

Main article: Jones polynomial
The Jones polynomial drawn in the Tutte plane

Along the hyperbola xy=1, the Tutte polynomial specialises to the Jones polynomial of an alternating knot if G is planar.

Individual points[edit]

(2,1)[edit]

T_G(2,1) counts the number of forests, i.e., the number of acyclic edge subsets.

(1,1)[edit]

T_G(1,1) counts the number of spanning forests (edge subsets without cycles and the same number of connected components as G).

(1,2)[edit]

T_G(1,2) counts the number of spanning subgraphs (edge subsets with the same number of connected components as G).

(2,0)[edit]

T_G(2,0) counts the number of acyclic orientations of G.[8]

(0,2)[edit]

T_G(0,2) counts the number of strongly connected orientations of G.[9]

(0,−2)[edit]

If G is a 4-regular graph, then

(-1)^{|V|+k(G)}T_G(0,-2)

counts the number of Eulerian orientations of G. Here k(G) is the number of connected components of G.[8]

(3,3)[edit]

If G is the m × n grid graph, then 2 T_G(3,3) counts the number of ways to tile a rectangle of width 4m and height 4n with T-tetrominoes.[10]

If G is a planar graph, then 2 T_G(3,3) equals the sum over weighted Eulerian orientations in a medial graph of G, where the weight of an orientation is 2 to the number of saddle vertices of the orientation (that is, the number of vertices with incident edges cyclicly ordered "in, out, in out").[11]

Potts and Ising models[edit]

Main articles: Ising model and Potts model
The partition functions for the Ising model and the 3- and 4-state Potts models drawn in the Tutte plane.

Define the hyperbola in the xy−plane:

 H_2: \quad (x-1)(y-1)=2,

the Tutte polynomial specialises to the partition function, Z(\cdot), of the Ising model studied in statistical physics. Specifically, along the hyperbola H_2 the two are related by the equation:[12]

Z(G) = 2\left(e^{-\alpha}\right)^{|E| - r(E)} \left(4 \sinh \alpha \right )^{r(E)}  T_G \left (\coth \alpha, e^{2 \alpha} \right).

In particular,

(\coth \alpha - 1) \left(e^{2 \alpha} - 1 \right ) = 2

for all complex α.

More generally, if for any positive integer q, we define the hyperbola:

H_q: \quad (x-1)(y-1)=q,

then the Tutte polynomial specialises to the partition function of the q-state Potts model. Various physical quantities analysed in the framework of the Potts model translate to specific parts of the H_q.

Correspondences between the Potts model and the Tutte plane [13]
Potts model Tutte polynomial
Ferromagnetic Positive branch of H_q
Antiferromagnetic Negative branch of H_q with y>0
High temperature Asymptote of H_q to y=1
Low temperature ferromagnetic Positive branch of H_q asymptotic to x=1
Zero temperature antiferromagnetic Graph q-colouring, i.e., x=1-q, y=0

Flow polynomial[edit]

Main article: Nowhere-zero flow
The flow polynomial drawn in the Tutte plane

At x=0, the Tutte polynomial specialises to the flow polynomial studied in combinatorics. For a connected and undirected graph G and integer k, a nowhere-zero k-flow is an assignment of “flow” values 1,2,\dots,k-1 to the edges of an arbitrary orientation of G such that the total flow entering and leaving each vertex is congruent modulo k. The flow polynomial C_G(k) denotes the number of nowhere-zero k-flows. This value is intimately connected with the chromatic polynomial, in fact, if G is a planar graph, the chromatic polynomial of G is equivalent to the flow polynomial of its dual graph G^* in the sense that

Theorem (Tutte).

C_G(k)=k^{-1} \chi_{G^*}(k).

The connection to the Tutte polynomial is given by:

C_G(k)= (-1)^{|E|+|V|+k(G)} T_G(0,1-k).

Reliability polynomial[edit]

The reliability polynomial drawn in the Tutte plane

At x=1, the Tutte polynomial specialises to the all-terminal reliability polynomial studied in network theory. For a connected graph G remove every edge with probability p; this models a network subject to random edge failures. Then the reliability polynomial is a function R_G(p), a polynomial in p, that gives the probability that every pair of vertices in G remains connected after the edges fail. The connection to the Tutte polynomial is given by

R_G(p) = (1-p)^{|V|-k(G)} p^{|E|-|V|+k(G)} T_G \left (1, \tfrac{1}{p} \right).

Dichromatic polynomial[edit]

Tutte also defined a closer 2-variable generalization of the chromatic polynomial, the dichromatic polynomial of a graph. This is

Q_G(u,v) = \sum\nolimits_{A \subseteq E} u^{k(A)} v^{|A|-|V|+k(A)},

where k(A) is the number of connected components of the spanning subgraph (V,A). This is related to the corank-nullity polynomial by

Q_G(u,v) = u^{k(G)} \, R_G(u,v).

The dichromatic polynomial does not generalize to matroids because k(A) is not a matroid property: different graphs with the same matroid can have different numbers of connected components.

Related polynomials[edit]

Martin polynomial[edit]

Main article: Martin polynomial

The Martin polynomial m_{\vec{G}}(x) of an oriented 4-regular graph \vec{G} was defined by Pierre Martin in his 1977 thesis.[14] In this work, Martin showed that if G is a plane graph and \vec{G}_m is its directed medial graph, then

T_G(x,x) = m_{\vec{G}_m}(x).

Algorithms[edit]

Deletion–contraction[edit]

The deletion–contraction algorithm applied to the diamond graph. Red edges are deleted in the left child and contracted in the right child. The resulting polynomial is the sum of the monomials at the leaves, x^3+2x^2 +y^2+2xy+x+y. Based on Welsh & Merino (2000).

The deletion–contraction recurrence for the Tutte polynomial,

T_G(x,y)= T_{G \setminus e}(x,y) + T_{G/e}(x,y), \qquad e \text{ not a loop nor a bridge.}

immediately yields a recursive algorithm for computing it: choose any such edge e and repeatedly apply the formula until all edges are either loops or bridges; the resulting base cases at the bottom of the evaluation are easy to compute.

Within a polynomial factor, the running time t of this algorithm can be expressed in terms of the number of vertices n and the number of edges m of the graph,

t(n+m)= t(n+m-1) + t(n+m-2),

a recurrence relation that scales as the Fibonacci numbers with solution[15]

 t(n+m)= \left (\frac{1+\sqrt{5}}{2} \right )^{n+m} = O \left (1.6180^{n+m} \right ).

The analysis can be improved to within a polynomial factor of the number \tau(G) of spanning trees of the input graph.[16] For sparse graphs with m=O(n) this running time is O(\exp(n)). For regular graphs of degree k, the number of spanning trees can be bounded by

\tau(G) = O \left (\nu_k^n n^{-1} \log n \right ),

where

\nu_k = \frac{(k-1)^{k-1}}{(k^2-2k)^{\frac{k}{2}-1}}.

so the deletion–contraction algorithm runs within a polynomial factor of this bound. For example:[17]

\nu_5 \approx 4.4066.

In practice, graph isomorphism testing is used to avoid some recursive calls. This approach works well for graphs that are quite sparse and exhibit many symmetries; the performance of the algorithm depends on the heuristic used to pick the edge e.[18]

Gaussian elimination[edit]

In some restricted instances, the Tutte polynomial can be computed in polynomial time, ultimately because Gaussian elimination efficiently computes the matrix operations determinant and Pfaffian. These algorithms are themselves important results from algebraic graph theory and statistical mechanics.

T_G(1,1) equals the number \tau(G) of spanning trees of a connected graph. This is computable in polynomial time as the determinant of a maximal principal submatrix of the Laplacian matrix of G, an early result in algebraic graph theory known as Kirchhoff’s Matrix–Tree theorem. Likewise, the dimension of the bicycle space at T_G(-1,-1) can be computed in polynomial time by Gaussian elimination.

For planar graphs, the partition function of the Ising model, i.e., the Tutte polynomial at the hyperbola H_2, can be expressed as a Pfaffian and computed efficiently via the FKT algorithm. This idea was developed by Fisher, Kasteleyn, and Temperley to compute the number of dimer covers of a planar lattice model.

Markov chain Monte Carlo[edit]

Using a Markov chain Monte Carlo method, the Tutte polynomial can be arbitrarily well approximated along the positive branch of H_2, equivalently, the partition function of the ferromagnetic Ising model. This exploits the close connection between the Ising model and the problem of counting matchings in a graph. The idea behind this celebrated result of Jerrum and Sinclair[19] is to set up a Markov chain whose states are the matchings of the input graph. The transitions are defined by choosing edges at random and modifying the matching accordingly. The resulting Markov chain is rapidly mixing and leads to “sufficiently random” matchings, which can be used to recover the partition function using random sampling. The resulting algorithm is a fully polynomial-time randomized approximation scheme (fpras).

Computational complexity[edit]

Several computational problems are associated with the Tutte polynomial. The most straightforward one is

Input
A graph G
Output
The coefficients of T_G

In particular, the output allows evaluating T_G(-2,0) which is equivalent to counting the number of 3-colourings of G. This latter question is #P-complete, even when restricted to the family of planar graphs, so the problem of computing the coefficients of the Tutte polynomial for a given graph is #P-hard even for planar graphs.

Much more attention has been given to the family of problems called Tutte(x,y) defined for every complex pair (x,y):

Input
A graph G
Output
The value of T_G(x,y)

The hardness of these problems varies with the coordinates (x,y).

Exact computation[edit]

The Tutte plane. Every point (x,y) in the real plane corresponds to a computational problem T_G(x,y). At any red point, the problem is polynomial-time computable; at any blue point, the problem is #P-hard in general, but polynomial-time computable for planar graphs; and at any point in the white regions, the problem is #P-hard even for bipartite planar graphs.

If both x and y are non-negative integers, the problem T_G(x,y) belongs to #P. For general integer pairs, the Tutte polynomial contains negative terms, which places the problem in the complexity class GapP, the closure of #P under subtraction. To accommodate rational coordinates (x,y), one can define a rational analogue of #P.[20]

The computational complexity of exactly computing T_G(x,y) falls into one of two classes for any x, y \in \mathbb{C}. The problem is #P-hard unless (x,y) lies on the hyperbola H_1 or is one of the points

\left \{ (1,1), (-1,-1), (0,-1), (-1,0), (i,-i), (-i,i), \left(j,j^2 \right), \left(j^2,j \right) \right \}, \qquad j = e^{\frac{2 \pi i}{3}}.

In which cases it is computable in polynomial time.[21] If the problem is restricted to the class of planar graphs, the points on the hyperbola H_2 become polynomial-time computable as well. All other points remain #P-hard, even for bipartite planar graphs.[22] In his paper on the dichotomy for planar graphs, Vertigan claims (in his conclusion) that the same result holds when further restricted to graphs with vertex degree at most three, save for the point T_G(0,-2), which counts nowhere-zero Z3-flows and is computable in polynomial time.[23]

These results contain several notable special cases. For example, the problem of computing the partition function of the Ising model is #P-hard in general, even though celebrated algorithms of Onsager and Fisher solve it for planar lattices. Also, the Jones polynomial is #P-hard to compute. Finally, computing the number of four-colourings of a planar graph is #P-complete, even though the decision problem is trivial by the four colour theorem. In contrast, it is easy to see that counting the number of three-colourings for planar graphs is #P-complete because the decision problem is known to be NP-complete via a parsimonious reduction.

Approximation[edit]

The question which points admit a good approximation algorithm has been very well studied. Apart from the points that can be computed exactly in polynomial time, the only approximation algorithm known for T_G(x,y) is Jerrum and Sinclair’s FPRAS, which works for points on the “Ising” hyperbola H_2 for y > 0. If the input graphs are restricted to dense instances, with degree \Omega(n), there is an FPRAS if x ≥ 1, y ≥ 1.[24]

Even though the situation is not as well understood as for exact computation, large areas of the plane are known to be hard to approximate.[20]

See also[edit]

Bollobás–Riordan polynomial

Notes[edit]

  1. ^ Chap. 10 in Bollobás (1998), chap. 13 in Biggs (1993), chap. 15 in Godsil & Royle (2004).
  2. ^ cf. Sokal (2005)
  3. ^ eq. (2.26) in Sokal (2005)
  4. ^ a b Tutte (2004)
  5. ^ Welsh
  6. ^ See Farr (2007)
  7. ^ Farr (2007)
  8. ^ a b Welsh (1999)
  9. ^ Las Vergnas (1980).
  10. ^ Korn & Pak (2004), see also Korn & Pak (2003) for combinatorial interpretations of many other points
  11. ^ Las Vergnas (1988)
  12. ^ Welsh, Dominic (1993). Complexity: Knots, Colourings and Counting. London Mathematical Society Lecture Note Series. Cambridge University Press. p. 62. ISBN 978-0521457408. 
  13. ^ Welsh & Merino (2000)
  14. ^ Martin, Pierre (1977). Enumérations Eulériennes dans les multigraphes et invariants de Tutte-Grothendieck [Eulerian Enumerations in multigraphs and Tutte-Grothendieck invariants] (Thesis) (in French). Joseph Fourier University. 
  15. ^ Wilf (1986)
  16. ^ Sekine, Imai & Tani (1995)
  17. ^ Chung & Yau (1999), following Björklund et al. (2008)
  18. ^ Sekine, Imai & Tani (1995), Imai (2000), Haggard, Pierce & Royle (2008)
  19. ^ Jerrum & Sinclair (1993)
  20. ^ a b Goldberg & Jerrum (2008)
  21. ^ Jaeger, Vertigan & Welsh (1990)
  22. ^ Vertigan & Welsh (1992)
  23. ^ Vertigan, Dirk (2005). "The Computational Complexity of Tutte Invariants for Planar Graphs". SIAM J. Comput. 35 (3): 690–712. doi:10.1137/S0097539704446797. 
  24. ^ x ≥ 1, y = 1 is given by Annan (1994). x ≥ 1, y > 1 is given by Alon, Frieze & Welsh (1995)

References[edit]

External links[edit]