Degree Associated Reconstruction Number of Graphs with Regular Pruned Graph

Abstract

A vertex-deleted unlabeled subgraph of a graph \(G\) is called a card of \(G\). A card of \(G\) with which the degree of the deleted vertex is also given is called a degree-associated card or dacard of \(G\). The degree-associated reconstruction number, \(\mathrm{drn}(G)\), of a graph \(G\) is the size of the smallest collection of dacards of \(G\) that uniquely determines \(G\). The maximal subgraph without end vertices of a graph \(G\) that is not a tree is called the pruned graph of \(G\). It is shown that \(\mathrm{drn}\) of some connected graphs with regular pruned graph is \(2\) or \(3\).