Some Results on 2-Vertex Switching in Joints - Jayasekaran, C.; Sudha, J. Christabel; Shijo, M. Ashwin (Communications in Mathematics and Applications, 2021)
For a finite undirected graph \(G(V,E)\) and a non empty subset \(σV\), the switching of \(G\) by \(σ\) is defined as the graph \(G^σ(V,E’)\) which is obtained from \(G\) by removing all edges between \(σ\) and its complement \(V\)-\(σ\) and adding as edges all non-edges between \(σ\) and \(V\)-\(σ\). For \(σ = {v}\), we write \(G^v\) instead of \(G^{v \) and the corresponding switching is called as vertex switching. We also call it as \(|σ |\)-vertex switching. When \(|σ | = 2\), we call it as 2-vertex switching. A subgraph \(B\) of \(G\) which contains \(G[σ ]\) is called a joint at \(σ\) in \(G\) if \(B\)-\(σ\) 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 \(σ = {u,v}\) in \(G\) to be a \(c\)-joint and a \(d\)-joint at \(σ\) in \(G^σ\) and also a necessary and sufficient condition for a \(d\)-joint \(B\) at \(σ = {u,v}\) in \(G\) to be a \(c\)-joint and a \(d\)-joint at \(σ\) in \(G^σ\) when \(uvE(G)\) and when \(uv otin E(G)\).