On Constant-Size Graphs That Preserve the Local Structure of High-Girth Graphs

Let G=(V,E) be an undirected graph with maximum degree d. The k-disc of a vertex v is defined as the rooted subgraph that is induced by all vertices whose distance to v is at most k. The k-disc frequency vector of G, freq(G), is a vector indexed by all isomorphism types of k-discs. For each such isomorphism type Gamma, the k-disc frequency vector counts the fraction of vertices that have k-disc isomorphic to Gamma. Thus, the frequency vector freq(G) of G captures the local structure of G. A natural question is whether one can construct a much smaller graph H such that H has a similar local structure. N. Alon proved that for any epsilon>0 there always exists a graph H whose size is independent of |V| and whose frequency vector satisfies ||freq(G) - freq(G)||_1 <= epsilon. However, his proof is only existential and neither gives an explicit bound on the size of H nor an efficient algorithm. He gave the open problem to find such explicit bounds. In this paper, we solve this problem for the special case of high girth graphs. We show how to efficiently compute a graph H with the above properties when G has girth at least 2k+2 and we give explicit bounds on the size of H.