AbstractWe observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601–622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39–54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276–290], using a different algorithm based on logical techniques

Traldi, Lorenzo

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Discrete Applied Mathematics 154 (2006) 1032–1036
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Note
On the colored Tutte polynomial of a graph of bounded treewidth
Lorenzo Traldi
Department of Mathematics, Lafayette College, Easton, PA 18042, USA
Received 6 February 2003; received in revised form 9 September 2005; accepted 27 September 2005
Available online 2 December 2005
Abstract
We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601–622] for
the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak
[An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39–54] may be directly adapted
to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by
Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005)
276–290], using a different algorithm based on logical techniques.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Colored Tutte polynomial; k-sum; Splitting formula; Treewidth
1. Introduction
The Tutte polynomial (or dichromate) is one of the most-studied invariants of graphs and matroids; it has many
important applications, including vertex colorings, ﬂows, reliability, polynomial invariants of knots, and partition
functions of models of statistical mechanics. We refer the reader to [5,21,23] for detailed expositions. Many of these
applications involve some kind of edge-weighting or edge-coloring; for instance in knot theory an edge of a plane graph
represents a crossing of a knot diagram, and is colored to indicate which of the two possible crossings is present.
In our discussion we follow standard terminology of graph theory and matroid theory, as for instance in [5,17,22].
A graph G has an associated cycle matroid M, whose rank function is deﬁned as follows. For a subset S ⊆ E = E(G)
let G : S be the subgraph of G with V (G : S) = V (G) and E(G : S) = S; if c(G : S) is the number of connected
components of G : S then r(S) = |V (G)| − c(G : S). Whenever we discuss matroids, we presume that graphs are
included in the discussion via their cycle matroids.
The (uncolored)Tutte polynomialmay be deﬁned in several equivalentways.One is a deletion–contraction recursion:
if M is a matroid then T (M) is an element of the polynomial ring Z[X, Y ] such that T (M)=XT (M/e) for any isthmus
e of M, T (M) = YT (M − e) for any loop e of M, and T (M) = T (M − e) + T (M/e) for any other element e. The
empty matroid ∅ has T (∅) = 1. Another deﬁnition utilizes a subset expansion:
T (M) =
∑
S⊆E
(X − 1)r(E)−r(S)(Y − 1)|S|−r(S).
E-mail address: traldil@lafayette.edu.
0166-218X/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2005.09.013
L. Traldi / Discrete Applied Mathematics 154 (2006) 1032–1036 1033
Here r denotes the rank function of the matroid M. A third deﬁnition of the Tutte polynomial is phrased in terms of
basis activities; this deﬁnition will not be useful in this note, so we do not discuss it in detail.
The evident exponential character of these deﬁnitions suggests the result of [10] that computing T (M) is #P -hard
in general.
Pathwidth and treewidth were introduced in [18–20] as measures of the structural complexity of graphs, and since
then these notions have proven to be of great interest. Many algorithmically intractable problems become tractable
when restricted to graphs of bounded treewidth; in particular, the result that computing T (M) is tractable for graphs
of bounded treewidth was mentioned by Welsh in [21], and proven by Andrzejak in [2]. (See also [15].) The crucial
ingredients of Andrzejak’s proof are algorithms of Bodlaender and Hagerup which ﬁnd a binary tree decomposition
of a graph G of treewidth bounded by k [3,4] and Negami’s splitting formula for the Tutte polynomial of the k-sum
of two graphs [14]; the splitting formula is used repeatedly as G is constructed from simpler graphs according to a
binary tree decomposition. Makowsky has recently proven a broad generalization of this result [11–13], using logical
techniques; this generalization covers colored versions of the Tutte polynomial and many other kinds of generating
functions deﬁnable in monadic second order logic.
In this note we observe that Negami’s splitting formula can be extended to a colored version of the Tutte poly-
nomial, and hence that for this polynomial and its evaluations (like the Kauffman bracket) a colored version of
Andrzejak’s algorithm may be used. As Makowsky observes in [13], the greater generality of logical techniques
comes at a signiﬁcant computational price; an algorithm based on Negami’s splitting formula is less general but more
practical.
2. Negami’s splitting formula, with colors
Suppose a matroid M on a set E is colored by a function c : E → , and suppose that for each  ∈  elements
x, y of a ﬁeld have been chosen. (We trust that there will be no confusion between the color c(e) of an edge of G and
the number of components c(H) of a subgraph of G.) Zaslavsky [24] calls
R(M, c) =
∑
S⊆E
(∏
e∈S
xc(e)
)( ∏
e∈E−S
yc(e)
)
(X − 1)r(E)−r(S)(Y − 1)|S|−r(S)
the normal function of the colored matroid. R(M, c) is probably the most natural colored version of T (M), though
there are many others; see [6,9,24] and the references mentioned there. R(M, c) has a basis activities expansion
analogous to that of T (M), and it satisﬁes a colored deletion–contraction recursion: R(M, c) = ((X − 1)yc(e) +
xc(e))R(M/e, c) for any isthmus e of M, R(M, c) = ((Y − 1)xc(e) + yc(e))R(M − e, c) for any loop e of M, and
R(M, c)=yc(e)R(M−e, c)+xc(e)R(M/e, c) for any other element e. The empty matroid ∅ hasR(∅, c)=1. Zaslavsky
proves that R(M, c) is the most interesting ﬁeld-valued invariant which satisﬁes such a weighted deletion–contraction
recursion; any other such invariant is either degenerate (determined by the setE, the rank ofM and the loops and coloops
of M) or non-global (deﬁned only for certain choices of the x and y). This result generalizes a universal property
of T (M) [8].
It may be surprising that R(M, c) can be a simpler invariant to understand than T (M). If we regard T (M) as a
subset expansion, it is a generating function for the number of subsets S of E with given corank r(E)− r(S) and nullity
|S|− r(S). Similarly, if the various x and y are independent indeterminates then R(M, c) is a generating function for
the number of subsets S of E with given corank r(E) − r(S), nullity |S| − r(S), and distribution of colored elements
inside and outside S. Why is this simpler? Well, if the function c : E →  is injective then the distribution of colored
elements inside and outside S completely determines S. After determining r(E) as the corank of ∅, we see that if c is
injective and the various x and y are independent indeterminates then R(M, c) is essentially a list which gives the
rank of every S ⊆ E, i.e., R(M, c) is essentially the rank function of M.
Theorem 2.1. Let G be a k-sum of subgraphs H and K, i.e., V (G) = V (H) ∪ V (K), |V (H) ∩ V (K)| = k, E(G) =
E(H)∪E(K) and E(H)∩E(K)=∅. Then there is a formula which depends only on k, |V (G)| and c(G) which gives
R(M, c) as a function of the R-invariants and numbers of connected components of contractions of H and K.
1034 L. Traldi / Discrete Applied Mathematics 154 (2006) 1032–1036
The proof of Theorem 2.1 follows the proof of Theorem 4.2 in [14] very closely. The key to what Negami calls his
“beautiful formula” is to study not R(M, c) but a related invariant f (G, c) with a simple deletion–contraction property
which treats loops, isthmuses and other edges in the same way.
Let G be a graph, and suppose E = E(G) is colored by c : E → . Suppose that t is an element of a ﬁeld and for
each  ∈ , x and y are also elements of the ﬁeld; for instance, the ﬁeld in question could be the ﬁeld of quotients
of the polynomial ring Z[{t} ∪ {x, y :  ∈ }]. Then the deletion–contraction formula f (G, c) = xc(e)f (G/e, c) +
yc(e)f (G − e, c), together with the initial condition that if E(G) = ∅ then f (G, c) = t |V (G)|, yields a well-deﬁned
invariant f (G, c).
Proposition 2.2.
f (G, c) =
∑
S⊆E(G)
(∏
e∈S
xc(e)
)( ∏
e∈E−S
yc(e)
)
tc(G:S).
Proof. This is true by deﬁnition if E(G) = ∅. Suppose inductively that e0 ∈ E(G); then
f (G, c) = xc(e0)f (G/e0, c) + yc(e0)f (G − e0, c)
= xc(e0) ·
∑
S⊆E(G/e0)
(∏
e∈S
xc(e)
)⎛⎝ ∏
e∈E−{e0}−S
yc(e)
⎞
⎠ tc((G/e0):S)
+ yc(e0) ·
∑
S⊆E(G−e0)
(∏
e∈S
xc(e)
)⎛⎝ ∏
e∈E−{e0}−S
yc(e)
⎞
⎠ tc((G−e0):S)
=
∑
e0∈S⊆E(G)
(∏
e∈S
xc(e)
)( ∏
e∈E−S
yc(e)
)
tc(G:S)
+
∑
e0 /∈S⊆E(G)
(∏
e∈S
xc(e)
)( ∏
e∈E−S
yc(e)
)
tc(G:S). 
It follows from Proposition 2.2 that f (G, c) is closely related to R(M, c); the corresponding relationship between
the uncolored invariants f (G) and T (M) is well known [14,16].
Corollary 2.3. (X − 1)c(G)(Y − 1)|V (G)|R(M, c) is obtained from f (G, c) by evaluating t → (X − 1)(Y − 1), each
xc(e) → xc(e)(Y − 1) and each yc(e) → yc(e). Also, t−c(G)f (G, c) is obtained from R(M, c) by evaluating X → t + 1,
Y → 2, each xc(e) → xc(e) and each yc(e) → yc(e).
Suppose G is a k-sum of subgraphs H and K, i.e., V (G)=V (H)∪V (K), |V (H)∩V (K)|=k,E(G)=E(H)∪E(K)
and E(H) ∩ E(K) = ∅. Let U = V (H) ∩ V (K), and let (U) be the set of all partitions of U; (U) is a lattice when
ordered by reﬁnement. For , ′ ∈ (U) let ∧ ′ be the meet in this lattice. That is, ∧ ′ is the partition deﬁned by: u
and u′ are in the same element of ∧ ′ if there is a sequence u = u1, u2, . . . , ua = u′ such that when 1 i < a, ui and
ui+1 are in the same element of either  or ′. Using an arbitrary ordering of (U) as {1, 2, . . .} Negami [14] deﬁnes
Tk to be the square matrix whose (i, j) entry is t |i∧j | and observes that if t is an indeterminate then Tk is nonsingular:
the product of terms on the diagonal is clearly of higher degree than any other product of terms that contributes to the
determinant, so the determinant must be nonzero.
For  ∈ (U) a contraction G/may be deﬁned in the obvious way: each element of  gives rise to a single vertex of
G/ and each edge e of G gives rise to an edge e of G/which connects the vertices of G/ containing the end-vertices
of e. Clearly non-loop edges of G may become loops in G/, and isthmuses of G may become non-isthmuses in G/.
Proposition 2.4 gives a splitting formula for f (G, c); Theorem 2.1 follows immediately from this formula and
Corollary 2.3. Observe that |V (G)| and the various connected component counts which appear in Theorem 2.1 come
from Corollary 2.3, not Proposition 2.4; in the terminology of [14] f (G, c) splits nicely but R(M, c) does not. We do
not give the proof of Proposition 2.4 because it is no different from Negami’s, but it is worth noting that the argument
L. Traldi / Discrete Applied Mathematics 154 (2006) 1032–1036 1035
depends on the fact that the deletion–contraction property of f is insensitive to whether an edge is a loop, an isthmus,
or neither; this insensitivity makes it easy to compare computations of f (G, c) and f (H, c).
Proposition 2.4. Let t f(K/(U), c) be the row vector (f (K/1, c), f (K/2, c), . . .) and f(H/(U), c) be the column
vector⎛
⎝f (H/1, c)f (H/2, c)
...
⎞
⎠
.
Then
t f(K/(U), c) · T −1k · f(H/(U), c) = f (G, c).
Corollary 2.5. If G is a colored graph of bounded treewidth, R(M, c) may be calculated in polynomial time.
Andrzejak [2] gives a detailed discussion of an algorithm which calculates T (M) by determining a binary, rooted
tree decomposition of G and repeatedly using Negami’s splitting formula; the algorithm runs in time O(n2+7 log2 c),
where n = |V (G)| and c is twice the number of partitions of a set with 3k + 3 elements, presuming that arithmetic
operations are performed at constant cost. The same discussion applies here, with the caveat that the cost of arithmetic
operations may depend on the ﬁeld that contains the x and y.
Similarly, if we consider the relationship between R(M, c) and the rank function of M we conclude:
Corollary 2.6. For a graph G of bounded treewidth the rank function of the cycle matroid M may be determined in
polynomial time.
We do not know how these results may generalize to matroids which are not cycle matroids of graphs; Negami’s
argument requires both the simple deletion–contraction recursion of f and the presence of vertices in H ∩K , so it does
not apply to arbitrary matroids. There are results in the literature regarding Tutte polynomials of generalized parallel
connections of general matroids [1,7], but those generalized parallel connections are not sufﬁciently general to include
all k-sums of graphs.
Acknowledgements
We are indebted to J.A. Makowsky for pointing out the potential usefulness of a colored version of Negami’s splitting
formula, and for his interesting and instructive correspondence. We are also grateful to G. Gordon and the referees for
comments on early versions of this note.
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On the colored Tutte polynomial of a graph of bounded treewidth 

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On the colored Tutte polynomial of a graph of bounded treewidth

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