On Hanf-equivalence and the number of embeddings of small induced subgraphs

Two graphs are Hanf-equivalent with respect to radius r if there is a bijection between their vertex sets which preserves the isomorphism types of the vertices' neighbourhoods of radius r. For r = 1 this means that the graphs have the same degree sequence. In this paper we relate Hanf-equivalence to the graph-theoretical concept of subgraph equivalence. To make this concept applicable to graphs that are not necessarily connected, we first generalise the notion of the radius of a connected graph to general graphs in a suitable way, which we call the generalised radius. We say that two graphs G and H are subgraph-equivalent up to generalised radius r if for all graphs S of generalised radius r, the number of induced subgraphs isomorphic to S is the same in G and H. We prove that Hanf-equivalence with respect to radius r is equivalent to subgraph-equivalence up to generalised radius r, thereby relating the purely logical and the graph-theoretical concepts in a very strong way. The notion of subgraph-equivalence up to order s is defined accordingly, where all graphs S of order at most s are taken into account. As a corollary we obtain that Hanf-equivalence with respect to radius r implies subgraph-equivalence up to order s, provided that r ≥ 3s/4. In particular, this implies that two graphs which are Hanf-equivalent with respect to radius 3s/4 satisfy exactly the same unions of conjunctive queries of quantifier rank at most s.