Self Vertex Switchings of Trees

Abstract

A vertex $v\in V(G)$ is said to be a self vertex switching of $G$ if $G$ is isomorphic to $G^v$, where $G^v$ is the graph obtained from $G$ by deleting all edges of $G$ incident to $v$ and adding all edges incident to $v$ which are not in $G$. The set of all self vertex switchings of $G$ is denoted by $S{S}_1(G)$ and its cardinality by $s{s}_1(G)$. In [6], the number $s{s}_1(G)$ is calculated for the graphs cycle, path, regular graph, wheel, Euler graph, complete graph, and complete bipartite graphs. In this paper, for a vertex $v$ of a graph $G$, the graph $G^v$ is characterized for tree, star, and forest with a given number of components. Using this, we characterize trees and forests, each with a self vertex switching.