Reconstruction of bipartite graphs and triangle-free graphs with connectivity two

A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. The Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that all triangle-free graphs G with connectivity two such that diam(G) = 2 or diam(G) = diam(G) = 3 are reconstructible. We also prove that all 2-connected bipartite graphs G such that diam(G) = 2 or diam(G) = diam(G) = 3 are reconstructible and show that RC is true if and only if all 2-connected graphs H containing an odd cycle such that diam(H) = 2 or diam(H) = diam(H) = 3 are reconstructible.