The Local Structure of Bounded Degree Graphs

TL;DR

This paper investigates the problem of approximating the local structure of large bounded degree graphs with small, fixed-size graphs, proving the problem's undecidability in general and decidability for paths.

Contribution

It establishes the undecidability of constructing small graphs with similar local structure to large graphs, and provides explicit bounds for paths.

Findings

Undecidability of the problem for general graphs.

Decidability and explicit bounds for paths.

Extension of local structure approximation to directed edge-colored paths.

Abstract

Let $G=(V,E)$ be a simple graph with maximum degree $d$. For an integer $k\in\mathbb{N}$, the $k$-disc of a vertex $v\in V$ 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 $\text{freq}_{k}(G)$, is a vector indexed by all isomorphism types of rooted $k$-discs. For each such isomorphism type $\Gamma$, the corresponding entry in $\text{freq}_{k}(G)$ counts the fraction of vertices in $V$ that have a $k$-disc isomorphic to $\Gamma$. In a sense, $\text{freq}_{k}(G)$ 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 $\text{freq}_{k}(G)$ 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 $\epsilon>0$ there always exists a graph $H$ whose size is independent of $|V|$ and whose frequency vector satisfies $||\text{freq}_{k}(G)-\text{freq}_{k}(H)||_{1}\le\epsilon$. 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 $H$.