Reconstruction number of graphs with unique pendant vertex

Abstract

A vertex-deleted subgraph of a graph G is called a card of G. The reconstruction number of G is the minimum number of cards of G that suffices to determine G uniquely. A P-graph (Yongzhi, 1988) is a connected graph of order p with exactly two blocks, only one of them is K2 and the other block has a vertex of degree p−2 other than the cut vertex. It is shown that the reconstruction number of P-graphs is three in most of the cases, which strengthens the result of Bollobas (1990). Finally, we conclude that the reconstruction number of most of the separable graphs (connected graphs with a cut vertex) with pendant vertices is three if the reconstruction number of all other connected graphs without pendant vertices is three.