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Counting Homomorphisms to Square-Free Graphs, Modulo 2

Published:25 May 2016Publication History
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Abstract

We study the problem ⊕HomsToH 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 (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕HomsToH is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. 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.

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          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 8, Issue 3
          May 2016
          105 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/2936308
          Issue’s Table of Contents

          Copyright © 2016 ACM

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 25 May 2016
          • Accepted: 1 January 2016
          • Revised: 1 August 2015
          • Received: 1 May 2015
          Published in toct Volume 8, Issue 3

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