From a Zoo to a Zoology: Descriptive Complexity for Graph Polynomials

Abstract

We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce the class of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities.

On Counting Generalized Colorings

“…He has verified that all the examples of graph polynomials discussed in the literature, with the exception of the weighted graph polynomial of [NW99], are actually SOL-polynomials over some expansions (by adding order relations) of the graph, cf. also [Mak06]. In some cases this is straight forward, but in some cases it follows from intricate theorems.…”

“…An outline of such a study was presented in [Mak06]. In [Mak04] the second author has introduced the MSOL-definable and the SOL-definable graph polynomials, the class of graph polynomials where the range of summation is definable in (monadic) second order logic.…”

From Hilbert’s program to a logic tool box

“…In the period of 1980-1995 my work shifted to applications of logic to semantics of programming languages, logic programming and databases ( [Mak84,Mak92]), after which I shifted again (starting in 1995) to applications of logic to algorithmics and combinatorics, as documented in [Mak04,Mak06].…”

Graph Polynomials: From Recursive Definitions to Subset Expansion Formulas

“…Older graph polynomials, treated in monographs such as [Big93,God93,Bol99,GR01,Die05], are the characteristic polynomial, [CDS95], the chromatic polynomial, [DKT05], and the original Tutte polynomial, [Bol99]. A general program for the comparative study of graph polynomials was outlined in [Mak06,Mak07].…”