Some Results on 2-Vertex Switching in Joints

Abstract

For a finite undirected graph $G(V,E)$ and a non empty subset $\sigma \subseteq V$, the switching of $G$ by $\sigma$ is defined as the graph $G^{\sigma }(V,E^{\prime })$ which is obtained from $G$ by removing all edges between $\sigma$ and its complement $V$-$\sigma$ and adding as edges all non-edges between $\sigma$ and $V$-$\sigma$. For $\sigma =\{v\}$, we write $G^v$ instead of $G^{\{v\}}$ and the corresponding switching is called as vertex switching. We also call it as $|\sigma |$-vertex switching. When $|\sigma |=2$, we call it as 2-vertex switching.\ A subgraph $B$ of $G$ which contains $G[\sigma ]$ is called a joint at $\sigma$ in $G$ if $B$-$\sigma$ is connected and maximal. If $B$ is connected, then we call $B$ as $c$-joint otherwise $d$-joint. In this paper, we give a necessary and sufficient condition for a $c$-joint $B$ at $\sigma =\{u,v\}$ in $G$ to be a $c$-joint and a $d$-joint at $\sigma$ in $G^{\sigma }$ and also a necessary and sufficient condition for a $d$-joint $B$ at $\sigma =\{u,v\}$ in $G$ to be a $c$-joint and a $d$-joint at $\sigma$ in $G^{\sigma }$ when $uv\in E(G)$ and when $uv\notin E(G)$.