Branches and Joints in the Study of Self Switching of Graphs

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 $G$. Two vertices $u$ and $v$ in $G$ are said to be interchange similar if there exists an automorphism $\alpha$ of $G$ such that $\alpha (u)=v$ and $\alpha (v)=u$. In this paper, we give a characterization for a cut vertex in $G$ to be a self vertex switching where $G$ is a connected graph such that any two self vertex switchings, if they exist, are interchange similar.