1,671 research outputs found
The Circuit Partition Polynomial with Applications and Relation to the Tutte and Interlace Polynomials
This paper examines several polynomials related to the field of graph theory including the circuit partition polynomial, Tutte polynomial, and the interlace polynomial. We begin by explaining terminology and concepts that will be needed to understand the major results of the paper. Next, we focus on the circuit partition polynomial and its equivalent, the Martin polynomial. We examine the results of these polynomials and their application to the reconstruction of DNA sequences. Then we introduce the Tutte polynomial and its relation to the circuit partition polynomial. Finally, we discuss the interlace polynomial and its relationship to the Tutte and circuit partition polynomials
The Interlace Polynomial
In this paper, we survey results regarding the interlace polynomial of a
graph, connections to such graph polynomials as the Martin and Tutte
polynomials, and generalizations to the realms of isotropic systems and
delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials,
edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL
Weighted interlace polynomials
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend
to invariants of graphs with vertex weights, and these weighted interlace
polynomials have several novel properties. One novel property is a version of
the fundamental three-term formula
q(G)=q(G-a)+q(G^{ab}-b)+((x-1)^{2}-1)q(G^{ab}-a-b) that lacks the last term. It
follows that interlace polynomial computations can be represented by binary
trees rather than mixed binary-ternary trees. Binary computation trees provide
a description of that is analogous to the activities description of the
Tutte polynomial. If is a tree or forest then these "algorithmic
activities" are associated with a certain kind of independent set in . Three
other novel properties are weighted pendant-twin reductions, which involve
removing certain kinds of vertices from a graph and adjusting the weights of
the remaining vertices in such a way that the interlace polynomials are
unchanged. These reductions allow for smaller computation trees as they
eliminate some branches. If a graph can be completely analyzed using
pendant-twin reductions then its interlace polynomial can be calculated in
polynomial time. An intuitively pleasing property is that graphs which can be
constructed through graph substitutions have vertex-weighted interlace
polynomials which can be obtained through algebraic substitutions.Comment: 11 pages (v1); 20 pages (v2); 27 pages (v3); 26 pages (v4). Further
changes may be made before publication in Combinatorics, Probability and
Computin
Interlace Polynomials for Multimatroids and Delta-Matroids
We provide a unified framework in which the interlace polynomial and several
related graph polynomials are defined more generally for multimatroids and
delta-matroids. Using combinatorial properties of multimatroids rather than
graph-theoretical arguments, we find that various known results about these
polynomials, including their recursive relations, are both more efficiently and
more generally obtained. In addition, we obtain several interrelationships and
results for polynomials on multimatroids and delta-matroids that correspond to
new interrelationships and results for the corresponding graphs polynomials. As
a tool we prove the equivalence of tight 3-matroids and delta-matroids closed
under the operations of twist and loop complementation, called vf-safe
delta-matroids. This result is of independent interest and related to the
equivalence between tight 2-matroids and even delta-matroids observed by
Bouchet.Comment: 35 pages, 3 figure
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
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