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**1 - 1**of**1**### Tight Bounds for Graph Homomorphism and Subgraph Isomorphism

"... We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homo-morphism from graph G to graph H cannot be done in time |V (H)|o(|V (G)|). We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibili ..."

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We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homo-morphism from graph G to graph H cannot be done in time |V (H)|o(|V (G)|). We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of |V (H)|o(|V (H)|)-time algorithm deciding if graph G is a subgraph of H. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.