Graph Reconstruction from Connected Triples

TL;DR

This paper fully characterizes which graphs can be uniquely reconstructed from their connected triples, expanding understanding of graph reconstruction beyond previously known classes and identifying new classes where uniqueness holds or fails.

Contribution

It provides a complete characterization of graphs reconstructible from connected triples, including new classes where reconstruction is unique or not, advancing the theory of graph reconstruction from partial information.

Findings

Unique reconstruction for regular planar graphs and 5-connected planar graphs.

Non-uniqueness for k-connected planar graphs with k ≤ 4, Eulerian, and Hamiltonian graphs.

Extension of reconstruction results to various graph classes.

Abstract

The problem of graph reconstruction has been studied in its various forms over the years. In particular, the Reconstruction Conjecture, proposed by Ulam and Kelly in 1942, has attracted much research attention and yet remains one of the foremost unsolved problems in graph theory. Recently, Bastide, Cook, Erickson, Groenland, Kreveld, Mannens, and Vermeulen proposed a new model of partial information, where we are given the set of connected triples T_3, which is the set of 3-subsets of the vertex set that induce connected subgraphs. They proved that reconstruction is unique within the class of triangle-free graphs, 2-connected outerplanar graphs, and maximal planar graphs. They also showed that almost every graph can be uniquely reconstructed from their connected triples. However, little is known about other classes of non-triangle-free graphs within which reconstruction can occur uniquely, nor do we understand what kind of graphs can be uniquely reconstructed from their connected triples without assuming anything about the classes of graphs to which they belong. The main result of this paper is a complete characterization of all graphs that can be uniquely reconstructed from their connected triples T_3. We also show that reconstruction from T_3 is unique within the class of regular planar graphs, 5-connected planar graphs, certain strongly regular graphs, and complete multi-partite graphs, whereas it is not unique for the class of k-connected planar graphs with k less or equal to 4, Eulerian graphs, or Hamiltonian graphs.