Dichotomy Theorems for Homomorphism Polynomials of Graph Classes

Abstract: In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class, homomorphic to this $H$. These graphs will be encoded by a family of polynomials. We give dichotomies for the polynomials for cycles, cliques, trees, outerplanar graphs, planar graphs and graphs of bounded genus.