Dichotomy Theorems for Homomorphism Polynomials of Graph Classes

TL;DR

This paper establishes clear computational boundaries (dichotomies) for evaluating graph homomorphism polynomials across various graph classes within Valiant's algebraic model, clarifying which are efficiently computable and which are not.

Contribution

It introduces new dichotomy theorems for the complexity of graph homomorphism polynomials for multiple graph classes, expanding understanding of their computational complexity.

Findings

Dichotomy theorems for cycles, cliques, trees, outerplanar, planar, and bounded genus graphs.

Identification of classes with polynomial-time computable homomorphism polynomials.

Classification of graph classes where the problem is #P-hard.

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.