Exact Algorithms for Graph Homomorphisms

Abstract

Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During the recent years the topic of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can be extended to graph homomorphisms. By the celebrated result of Hell and Nešetřil, for each fixed simple graph H, deciding whether a given simple graph G has a homomorphism to H is polynomial-time solvable if H is a bipartite graph, and NP-complete otherwise. The case where H is a cycle of length 5 is the first NP-hard case different from graph coloring. We show that, for a given graph G on n vertices and an odd integer k≥ 5, whether G is homomorphic to a cycle of length k can be decided in time min\(\{(^{~n}_{n/k}), 2^{n/2}\} \cdot n^{\mathcal{O}(1)}\). We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows: If H is of treewidth at most t, then whether G is homomorphic to H can be decided in time \((2t + 1)^n \cdot n^{\mathcal{O}(1)}\).