Linear Recurrence Relations for Graph Polynomials

Abstract

A sequence of graphs G n is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let G n be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coefficients for the Tutte polynomials T(G i , x,y) = T(G i ), i.e.T(Gn + r) = p1(x,y) T(Gn + r − 1) + ... + p r (x,y) T(G n ).We show that such linear recurrences hold much more generally for a wide class of graph polynomials (also of labeled or signed graphs), namely they hold for all the extended MSOL-definable graph polynomials. These include most graph and knot polynomials studied in the literature.