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Volume 11 (2015) Article 2 pp. 35-57
The Complexity of Parity Graph Homomorphism: An Initial Investigation
Received: September 8, 2013
Revised: March 2, 2015
Published: March 14, 2015
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Keywords: complexity theory, graph homomorphisms, modular counting, dichotomy theorem
ACM Classification: F.2.2, G.3
AMS Classification: 68W25, 90C59, 68Q25, 68Q17

Abstract: [Plain Text Version]

$ \def\parityP{\oplus\mathrm{P}} $

Given a graph $G$, we investigate the problem of determining the parity of the number of homomorphisms from $G$ to some other fixed graph $H$. We conjecture that this problem exhibits a complexity dichotomy, such that all parity graph homomorphism problems are either polynomial-time solvable or $\parityP$--complete, and provide a conjectured characterisation of the easy cases.

We show that the conjecture is true for the restricted case in which the graph $H$ is a tree, and provide some tools that may be useful in further investigation into the parity graph homomorphism problem, and the problem of counting homomorphisms for other moduli.