2,111 research outputs found
A Tutte decomposition for matrices and bimatroids
AbstractWe develop a Tutte decomposition theory for matrices and their combinatorial abstractions, bimatroids. As in the graph or matroid case, this theory is based on a deletion–contraction decomposition. The contribution from the deletion, derived by an inclusion–exclusion argument, consists of three terms. With one more term contributed from the contraction, the decomposition has four terms in general. There are universal decomposition invariants, one of them being a corank–nullity polynomial. Under a simple change of variables, the corank–nullity polynomial equals a weighted characteristic polynomial. This gives an analog of an identity of Tutte. Applications to counting and critical problems on matrices and graphs are given
Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
We define 2-decompositions of ribbon graphs, which generalise 2-sums and
tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial
of such a 2-decomposition, and derive the classical Brylawski formula for the
Tutte polynomial of a tensor product as a (very) special case. This study was
initially motivated from knot theory, and we include an application of our
formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric
Graph polynomials and statistical physics
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (p. 53-54).We present several graph polynomials, of which the most important one is the Tutte polynomial. These various polynomials have important applications in combinatorics and statistical physics. We generalize the Tutte polynomial and establish its correlations to the other graph polynomials. Finally, our result about the decomposition of planar graphs and its application to the ice-type model is presented.by Jae Ill Kim.S.M
Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth
We consider the multivariate interlace polynomial introduced by Courcelle
(2008), which generalizes several interlace polynomials defined by Arratia,
Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present
an algorithm to evaluate the multivariate interlace polynomial of a graph with
n vertices given a tree decomposition of the graph of width k. The best
previously known result (Courcelle 2008) employs a general logical framework
and leads to an algorithm with running time f(k)*n, where f(k) is doubly
exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context
of tree decompositions, we give a faster and more direct algorithm. Our
algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently
implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor
improvements. 44 pages, 14 figure
The two-variable hypergraph Tutte polynomial via embedding activities
We prove that the two-variable Tutte polynomial of hypergraphs can be defined
via embedding activities. We also prove that embedding activities of
hypergraphs yield a Crapo-style decomposition of , thus
generalizing Bernardi's results from graphs to hypergraphs
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