A multivariate interlace polynomial

TL;DR

This paper introduces a new multivariate interlace polynomial that generalizes existing polynomials and demonstrates polynomial-time computability for certain graph classes, expanding the theoretical framework of graph polynomials.

Contribution

It defines a comprehensive multivariate interlace polynomial unifying previous variants and analyzes its computational complexity for graphs with bounded clique-width.

Findings

Generalizes several existing interlace polynomials

Polynomial-time evaluation for graphs of bounded clique-width

Full evaluation is exponential due to size constraints

Abstract

We define a multivariate polynomial that generalizes several interlace polynomials defined by Arratia, Bollobas and Sorkin on the one hand, and Aigner and van der Holst on the other. We follow the route traced by Sokal, who defined a multivariate generalization of Tutte's polynomial. We also show that bounded portions of our interlace polynomial can be evaluated in polynomial time for graphs of bounded clique-width. Its full evaluation is necessarly exponential just because of the size of the result.