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 \({SS_1}(G)\) and its cardinality by \(ss_1(G)\). In [6], the number \(ss_1(G)\) 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.