### Some Results on 2-Vertex Switching in Joints

C. Jayasekaran, J. Christabel Sudha, M. Ashwin Shijo

#### 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 $$|\sigma | = 2$$, 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 $$uv\notin E(G)$$.

#### Keywords

Switching; 2-vertex self switching; $$SS_2(G)$$; $$ss_2(G)$$

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#### References

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DOI: http://dx.doi.org/10.26713%2Fcma.v12i1.1426

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