Degree Associated Reconstruction Number of Split Graphs with Some Biregular Independent Set

Abstract

A vertex-deleted subgraph of a graph G with which the degree of the deleted vertex is given is called a degree associated card of G. The degree associated reconstruction number (or drn) of a graph G is the size of the smallest collection of the degree associated cards of G that uniquely determines G.  A split graph G is a graph in which the vertices can be partitioned into an independent set and a clique. We prove that the drn is 1 or 2 for all split graphs G of order at least seven in which all the vertices in the independent set have degrees r and s whose distinct degrees differ by at least two.