Tutte polynomial expansions for 2-separable graphs

Abstract

Let Ĝ be a graph obtained from a graph G with no loops or coloops by replacing each edge e= uw of G by a connected graph H e that has only the vertices u and w in common with the rest of Ĝ. Two mutually dual formulas are proved for the Tutte polynomial of Ĝ in terms of parameters of the graphs H e and (in the one case) flow polynomials of subgraphs of G or (in the other case) tension polynomials of contractions of G. This generalizes the results of Read and Whitehead on homeomorphism classes of graphs.