Reconstruction and edge reconstruction of triangle-free graphs

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 is an element of V(G)}. Let diam(G) and kappa(G) denote the diameter and the connectivity of a graph G, respectively, and let G(2) := {G : diam(G) = 2} and G(3) := {G : diam(G) = diam((G) over bar) = 3}. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in G(2) boolean OR G(3). Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph G in G(2) boolean OR G(3) with kappa(G) = 2. Moreover, they asked whether the result still holds if kappa(G) >= 3. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in G(2) boolean OR G(3) which contain triangles.) The case when G is an element of G(3) and kappa(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 G(2) with kappa(G) = 3. We also prove similar results about the Edge Reconstruction Conjecture. (c) 2023 Elsevier B.V. All rights reserved.