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

Revised: March 2, 2015

Published: March 14, 2015

**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.