Abstract

$ \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.

Highlights

  • 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

  • Graph homomorphism is a natural generalisation of graph colouring, in which the restrictions on adjacencies between colours can be more general than in the usual graph colouring problem

  • Ordinary graph colouring is the special case of homomorphisms into the complete graph

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Summary

Graph homomorphism

Graph homomorphism is a natural generalisation of graph colouring, in which the restrictions on adjacencies between colours can be more general than in the usual graph colouring problem. Ordinary graph colouring is the special case of homomorphisms into the complete graph. There are a number of computational problems of the form: given an instance (graph) G return some information about Hom(G, H). The goal is to classify the complexity of the computational problem in terms of the graph H. The complexity of the decision version of the graph homomorphism problem was completely classified by Hell and Nešetril in [14]. THE COMPLEXITY OF PARITY GRAPH HOMOMORPHISM: AN INITIAL INVESTIGATION the number of H-colourings is odd or even. (See Theorem 3.8.) Informally, ⊕P is the class of problems that can be expressed in terms of deciding the parity of the number of accepting computations of a non-deterministic Turing machine; see Section 2 for a precise definition.

Modular counting complexity
Completeness
Reduction by automorphisms
The Lovász vector of a graph
Pinning colours to vertices
Rooted graphs
Building gadgets
Involution-free trees
The reduction
A dichotomy for trees
Other graphs
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