Counting Homomorphisms to Square-Free Graphs, Modulo 2

TL;DR

This paper establishes a dichotomy for counting homomorphisms modulo 2 to square-free graphs, showing the problem is either polynomial-time solvable or P-complete, extending previous results to a broader class of graphs.

Contribution

It proves a conjecture that counting homomorphisms modulo 2 to graphs without 4-cycles is either in P or P-complete, for a wide class of graphs including those with unbounded treewidth.

Findings

Dichotomy theorem for square-free graphs

Extension of previous conjecture to graphs with unbounded treewidth

Identification of complexity boundary based on 4-cycle presence

Abstract

We study the problem HomsTo$H$ of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph $H$. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (non-modular) counting, so subtle dichotomy theorems can arise. We show the following dichotomy: for any $H$ that contains no 4-cycles, HomsTo$H$ is either in polynomial time or is $\oplus P$-complete. This confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs including graphs of unbounded treewidth. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.