On recognizing graphs by numbers of homomorphisms

Let Hom(G,H) be the number of homomorphisms from a graph G to a graph H. A well-known result of Lovász states that the function Hom(.,H) from all graphs uniquely determines the graph H upto isomorphism. We study this function restricted to smaller classes of graphs. We show that several natural classes (2-degenerated graphs and non-bipartite graphs with bounded chromatic number) are sufficient to recognize all graphs, and provide description of graph properties that are recognizable by other classes (graphs with bounded tree-width and clique-width). We consider simple undirected graphs without loops and multiple edges, unless specified otherwise. Let A be the class of all such graphs. Let Hom(G,H) be the number of homomorphisms from a graph G to a graph H . Sometimes we use the empty graph Z (without any vertices and edges). For each H , Hom(Z,H) = 1. Let G≤H be the class of graphs G that have a homomorphism to H , i.e., such that Hom(G,H) > 0. Lovász [1] has proved that the function Hom(., H) uniquely determines the graph H upto isomorphism, i.e., that if we know the number of isomorphisms from each graph in A to H , we can uniquely reconstruct the graph H . In fact, the proof of this statement implies that knowledge of Hom(., H) from all graphs on at most |V (H)| vertices is sufficient. We are interested in what graphs and graph properties can be recognized using smaller classes of graphs (independent on H). Supported as project 1M0545 by the Czech Ministry of Education