Graph Reconstruction with Connectivity Queries

TL;DR

This paper develops output-polynomial algorithms for reconstructing certain classes of graphs from their connected k-sets, proving unique reconstructibility for triangle-free graphs and analyzing bounded degree graphs.

Contribution

It introduces algorithms for reconstructing graphs from connected k-sets, characterizes unique reconstructibility of triangle-free graphs, and analyzes structural properties of bounded degree graphs.

Findings

Triangle-free graphs are uniquely reconstructible.

Graphs with bounded maximum degree share a common local structure.

Reconstruction problem is NP-hard in a variant with partial connectivity information.

Abstract

We study a problem of reconstruction of connected graphs where the input gives all subsets of size k that induce a connected subgraph. Originally introduced by Bastide et al. (WG 2023) for triples ($k=3$), this problem received comprehensive attention in their work, alongside a study by Qi, who provided a complete characterization of graphs uniquely reconstructible via their connected triples, i.e. no other graphs share the same set of connected triples. Our contribution consists in output-polynomial time algorithms that enumerate every triangle-free graph (resp. every graph with bounded maximum degree) that is consistent with a specified set of connected $k$-sets. Notably, we prove that triangle-free graphs are uniquely reconstructible, while graphs with bounded maximum degree that are consistent with the same $k$-sets share a substantial common structure, differing only locally. We suspect that the problem is NP-hard in general and provide a NP-hardness proof for a variant where the connectivity is specified for only some $k$-sets (with $k$ at least 4).