3-Vertex Duplication Self Switching of Graphs
Keywords:
Switching, self vertex switching, 3-vertex self switching, dupli- cation self switching.Abstract
For a graph ????(????, ????), duplication of a vertex ???? of a graph ???? produces new graph ????(????????) by adding a new vertex ????′ such that ????(????′) = ????(????). A k-vertex duplication of a graph ???? produces new graph ????((????1, ????2, … . , ????????)????) by adding ???? new vertices ????1′, ????2′, … . , ????????′ as the duplication of any ???? vertices ????1, ????2, … . , ???????? of ???? such that ????(???????? ′) = ????(???????? ). σ = {????1, ????2, … . , ????????} ⊆ V(G) is called a ????-vertex duplication self switching of graph ???? if ????(????????) ≅ ????(????????)????. The set of all ????-vertex duplication self switching of ???? is denoted by ????????????????(????) and the number of elements in the set is denoted by ????????????????(????). When ???? =3, it is called as 3-Vertex Duplication Self Switching. In this paper, we provide the necessary and sufficient conditions needed for a graph to be 3-vertex duplication self switching. We also find ????????????3(????) of path, cycle and complete graph.
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