Efficient computation of generalized Ising polynomials on graphs with fixed clique-width

TL;DR

This paper introduces a new class of graph polynomials definable in MSOL on hypergraph vocabularies and proves they can be computed efficiently in fixed-parameter polynomial time with respect to clique-width.

Contribution

It defines an infinite class of MSOL-definable hypergraph graph polynomials and proves their fixed-parameter polynomial-time computability.

Findings

New class of MSOL-definable hypergraph graph polynomials

Proved FPPT computability with respect to clique-width

Extended the scope of algorithmic meta-theorems for graph polynomials.

Abstract

Graph polynomials which are definable in Monadic Second Order Logic (MSOL) on the vocabulary of graphs are Fixed-Parameter Tractable (FPT) with respect to clique-width. In contrast, graph polynomials which are definable in MSOL on the vocabulary of hypergraphs are fixed-parameter tractable with respect to tree-width, but not necessarily with respect to clique width. No algorithmic meta-theorem is known for the computation of graph polynomials definable in MSOL on the vocabulary of hypergraphs with respect to clique-width. We define an infinite class of such graph polynomials extending the class of graph polynomials definable in MSOL on the vocabulary of graphs and prove that they are Fixed-Parameter Polynomial Time (FPPT) computable, i.e. that they can be computed in time $O(n^{f(k)})$, where $n$ is the number of vertices and $k$ is the clique-width.