Weighted interlace polynomials

TL;DR

This paper extends interlace polynomials to weighted graphs, introduces a simplified three-term recurrence, and develops reduction techniques enabling polynomial-time computation for certain graph classes.

Contribution

It introduces a novel three-term recurrence formula and weighted pendant-twin reductions, simplifying computation and enabling polynomial-time calculation for graphs reducible by these methods.

Findings

Weighted interlace polynomials have a new three-term recurrence property.

Pendant-twin reductions preserve interlace polynomials and reduce computational complexity.

Graphs constructed via substitutions have interlace polynomials obtainable through algebraic substitutions.

Abstract

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 $q(G)$ that is analogous to the activities description of the Tutte polynomial. If $G$ is a tree or forest then these "algorithmic activities" are associated with a certain kind of independent set in $G$. 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.