The Local Structure of Bounded Degree Graphs
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Let G=(V,E) be a simple graph with maximum degree d. For an integer k∈ℕ, the k-disc of a vertex v∈ V is defined as the rooted subgraph of G that is induced by all vertices whose distance to v is at most k. The k-disc frequency distribution vector of G, denoted by freq_k(G), is a vector indexed by all isomorphism types of rooted k-discs. For each such isomorphism type Γ, the corresponding entry in freq_k(G) counts the fraction of vertices in V that have a k-disc isomorphic to Γ. In a sense, freq_k(G) is one way to represent the "local structure" of G. The graph G can be arbitrarily large, and so a natural question is whether given freq_k(G) it is possible to construct a small graph H, whose size is independent of |V|, such that H has a similar local structure. N. Alon proved that for any ϵ>0 there always exists a graph H whose size is independent of |V| and whose frequency vector satisfies ||freq_k(G)-freq_k(H)||_1≤ϵ. However, his proof is only existential and does not imply that there is a deterministic algorithm to construct such a graph H. He gave the open problem of finding an explicit deterministic algorithm that finds H, or proving that no such algorithm exists. Our main result is that Alon's problem is undecidable if and only if a much more general problem (involving directed edges and edge colors) is undecidable. We also prove that both problems are decidable for the special case when G is a path. We show that the local structure of any directed edge-colored path G can be approximated by a suitable fixed-size directed edge-colored path H and we give explicit bound on the size of H.
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