Reconstruction and edge reconstruction of triangle-free graphs

Abstract

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs {G−v:v∈V(G)}. Let diam(G) and κ(G) denote the diameter and the connectivity of a graph G, respectively, and let G2:={G:diam(G)=2} and G3:={G:diam(G)=diam(G‾)=3}. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in G2∪G3. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph G in G2∪G3 with κ(G)=2. Moreover, they asked whether the result still holds if κ(G)≥3. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in G2∪G3 which contain triangles.) The case when G∈G3 and κ(G)≥3 was recently confirmed by Devi Priya and Monikandan. In this paper, we further show the Reconstruction Conjecture holds for every triangle-free graph G in G2 with κ(G)=3. We also prove similar results about the Edge Reconstruction Conjecture.