Reconstructing graphs and their connectivity using graphlets

TL;DR

This paper explores how graphlet degree distributions can be used to reconstruct graphs, proving new conditions under which graphs are uniquely determined by their subgraph distributions, thus advancing understanding of the reconstruction conjecture.

Contribution

It introduces methods using graphlet degree distributions to reconstruct graphs with specific properties and expands known conditions for graph reconstructibility.

Findings

Graphs with a unique almost-asymmetric vertex-deleted subgraph are reconstructible.

Graphs with a vertex of degree ≤ 2 or ≥ n-2 are reconstructible from their (<= n-1)-gdd.

Conditions on (<= 3)-gdd relate to graph connectivity and non-existence of certain gdd patterns.

Abstract

Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex. Graphlet degree distribution (gdd) of a graph is a matrix containing graphlet degree sequence for all vertices in the given graph. A long standing open problem called reconstruction conjecture (RC) asks whether the structure of a graph is uniquely determined by the multiset of its vertex-deleted subgraphs. Graphlet degree distribution up to size (n - 1), (<= n - 1)-gdd, gives more information to reconstruct the graph and we use it to reconstruct any graph having a unique almost-asymmetric vertex-deleted subgraph, where almost-asymmetric means that at most one automorphism orbit has size larger than one. Moreover, we prove that any graph containing a vertex-cut of size 1 or any graph of order n having a vertex with degree at most 2 or at least n-2 is reconstructible from its (<= n - 1)-gdd, which expands results shown in the standard RC. We also discuss the relation between gdd and graph connectivity and the conditions on (<= 3)-gdd, whose breaking means that no graph with such gdd exists.