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, $\mathrm{d}\mathrm{r}\mathrm{n}(G)$, 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 $\mathrm{d}\mathrm{r}\mathrm{n}$ of some connected graphs with regular pruned graph is $2$ or $3$.