Exact algorithm for graph homomorphism and locally injective graph homomorphism

•We describe a new algorithm for graph homomorphism problem.•We adapt our algorithm for the locally injective graph homomorphism problem.•We apply the algorithm to some special cases of the graph homomorphism problem.

For graphs G and H, a homomorphism from G to H   is a function φ:V(G)→V(H)φ:V(G)→V(H), which maps vertices adjacent in G to adjacent vertices of H. A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in H  . Many cases of graph homomorphism and locally injective graph homomorphism are NP-complete, so there is little hope to design polynomial-time algorithms for them. In this paper we present an algorithm for graph homomorphism and locally injective homomorphism working in time O⁎((b+2)|V(G)|)O⁎((b+2)|V(G)|), where b is the bandwidth of the complement of H.