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For graphs $G$ and $H$, a homomorphism from $G$ to $H$ is a function $\varphi\colon V(G) \to V(H)$, which maps vertices adjacent in $G$ to adjacent verticesof $H$. A homomorphism is locally injective if no two vertices with a commonneighbor are mapped to a single vertex in $H$. Many cases of graph homomorphismand locally injective graph homomorphism are NP-complete, so there is littlehope to design polynomial-time algorithms for them. In this paper we present analgorithm for graph homomorphism and locally injective homomorphism working intime $\mathcal{O}^*((b + 2)^{|V(G)|})$, where $b$ is the bandwidth of thecomplement of $H$.

Exact Algorithm for Graph Homomorphism and Locally Injective Graph Homomorphism