2-Vertex Self Switching of Trees

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 \(G^\sigma (V, E')\) 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 \(|\sigma|=2\), it is termed as 2-vertex switching. If \(G \cong G^\sigma\), then it is called self 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 a \(c\)-joint and otherwise a \(d\)-joint. A graph with no cycles is called an acyclic graph. A connected acyclic graph is called a tree. In this paper, we give necessary and sufficient conditions for a graph \(G\), for which \(G^{\sigma}\) at \(\sigma =\{u, v\}\) to be connected and acyclic when \(uv\in E(G)\) and \(uv \notin E(G)\). Using this, we characterize trees with a 2-vertex self switching.