The complexity of parity graph homomorphism: an initial investigation

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 ⊕P–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. ACM Classification: F.1.3, G.2.2 AMS Classification: 05C30, 05C60, 68Q17, 68Q25