Reconstructing graphs and their connectivity using graphlets

TL;DR

This work investigates reconstructing graphs from graphlet degree distributions ($gdd$), an approach that extends the classical Reconstruction Conjecture by rooting subgraphs at vertices. It establishes that, from the $(\le n-1)$-gdd, many graph properties and even entire graphs are reconstructible for significant classes (e.g., graphs with a vertex-cut of size 1, or with an extreme vertex degree), and it generalizes Kelly-style connectivity insights to $k$-vertex-connected graphs. The paper also contrasts graphlets with motifs, showing greater discriminative power of $gdd$, and analyzes uniqueness and realizability questions for $gdd$, including asymmetry-based reconstruction and the existence problem for given $gdd$. Overall, graphlet-based descriptions provide a powerful local-topology fingerprint with substantial, but not complete, promise for graph reconstruction and graph property inference. These results open paths to broader classes of reconstructible graphs and to algorithmic decidability questions for graphlet inventories.

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.

Figures (3)

• Figure 1: The orderings $\gamma$ of graphs $\mathcal{G}_{}$ and $\vartheta$ of graphlets $\mathcal{G'}_{}$ up to size $5$ given by Pržulj przulj2007. Under each graph $F$ there is a label $G_{\gamma(F)}$. Automorphism orbits are distinguished by different shades of black, white and gray. The number $\vartheta((F, r))$ assigned to a graphlet (F, r) is written next to some vertex from the automorphism orbit of $r$. Image source : L-GRAAL.
• Figure 2: Two graphs that cannot be distinguished by $\leq 4$ motif distribution, but can be distinguished by $\leq 4$-gdd, see Tables \ref{['t:motifs']} and \ref{['t:graphlets']}.
• Figure 3: The two graphs $G_1$ and $G_2$ are constracted by taking a path on $n-2$ vertices and ending it with a triangle or a fork respectively. The grey vertices then have the same gds containing graphlets isomorphic to a path rooted at its end for lengths $1, \dots, n-1$, all such graphlets being represented with multiplicity $1$ with the exception of length $n-1$, which occurs twice.

Theorems & Definitions (42)

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• Proposition 8