A characterization of 2-neighborhood degree list of diameter 2 graphs

Abstract

Let $N_2DL(v)$ denote the set of degrees of vertices at distance 2 from $v$. The $2$-neighborhood degree list of a graph is a listing of $N_2DL(v)$ for every vertex $v$. A degree restricted $2$-switch on edges $v_1v_2$ and $w_1w_2$, where $deg(v_1)=deg(w_1)$ and $deg(v_2)=deg(w_2)$, is the replacement of a pair of edges $v_1v_2$ and $w_1w_2$ by the edges $v_1w_2$ and $v_2w_1$ given that $v_1w_2$ and $v_2w_1$ did not appear in the graph originally. Let $G$ and $H$ be two graphs of diameter 2 on the same vertex set. We prove that $G$ and $H$ have the same $2$-neighborhood degree list if and only if $G$ can be transformed into $H$ by a sequence of degree restricted $2$-switches.