On the Complexity of the Interlace Polynomial

We consider the two-variable interlace polynomial introduced by Arratia, Bollobas and Sorkin (2004). We develop graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except on one line, where it is trivially polynomial time computable, and four lines, where the complexity is still open. This solves a problem posed by Arratia, Bollobas and Sorkin (2004). In particular, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at any point except at 0.Comment: 18 pages, 1 figure; new graph transformation (adding cycles) solves some unknown points, error in the statement of the inapproximability result fixed; a previous version has appeared in the proceedings of STACS 200

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

Interlace Polynomial of a Special Eulerian Graph

In a recent paper, Arratia, Bollobas and Sorkin discussed a graph polynomial defined recursively, which they call the interlace polynomial. There have been previous results on the interlace polynomials for special graphs, such as paths, cycles, and trees. Applications have been found in biology and other areas. In this research, I focus on the interlace polynomial of a special type of Eulerian graph, built from one cycle of size n and n cycle three graphs. I developed explicit formulas by implementing the toggling process to the graph. I further investigate the coefficients and special values of the interlace polynomial. Some of them can describe properties of the considered graph. Aigner and Holst also defined a new interlace polynomial, called the Q-interlace polynomial, recursively, which can tell other properties of the original graph. One immediate application of the Q-interlace polynomial is that a special value of it is the number of general induced subgraphs with an odd number of general perfect matchings. Thus by evaluating the Q-interlace polynomial at a specific value, we determine the number of general induced subrgaphs with an odd number of general perfect matchings of the considered Eulerian graph

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