### 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 \(G^{\sigma}(V,E')\) 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 \(uv\notin E(G)\).

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Y. Alavi, F. Bucklay, M. Shamula and S. Riuz, Highly irregular m-chromatic graphs, Discrete Mathematics 72(1-3) (1988), 3 – 13, DOI: 10.1016/0012-365X(88)90188-4.

S. Avadayappan and M. Bhuvaneshwari, More results on self Vertex switching, International Journal of Modern Sciences and Engineering Technology 1(3) (2014), 10 – 17, URL: https://nebula.wsimg.com/7d6ce9710b5f85f23acefb895e83abe6?AccessKeyId=D81D660734BCB585516F&disposition=0&alloworigin=1.

D. G. Corneil and R. A. Mathon (editors), Geometry and Combinatorics, Selected Works of J. J. Seidel, Academic Press, Boston (1991), URL: https://books.google.co.in/books?hl=en&lr=&id=brziBQAAQBAJ&oi=fnd&pg=PP1&ots=GpliQTCmZc&sig=YHPYtT_IvtQNsblCbtuUfCbC6C4&redir_esc=y#v=onepage&q&f=false.

C. Jayasekaran, Self vertex switchings of trees, Ars Combinatoria CXXVII (2016), 33 – 43, URL: https://www.researchgate.net/publication/307690970_Self_vertex_switchings_of_trees_Ars_Combinatoria_Vol127_pp33-43.

C. Jayasekaran, Self vertex switchings of disconnected unicyclic graphs, Ars Combinatoria CXXIX (2016), 51 – 62, URL: https://www.researchgate.net/publication/309174937_Self_Vertex_Switchings_of_Disconnected_Unicyclic_Graphs.

V. V. Kamalappan, J. P. Joseph and C. Jayasekaran, Branches and joints in the study of self switching of graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 67 (2008), 111 – 122, URL: https://www.researchgate.net/publication/268636390_Branches_and_joints_in_the_study_of_self_switching_of_graphs.

J. J. Seidel, A survey of two-graphs, in Atti Convegno Internazionale Teorie Combinatorie (Rome, Italy, September 3-15, 1973, Accademia Nazionale dei Lincei), Tomo I, pp. 481 – 511 (1976).

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

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