Results of Paired Domination of Some Special Graph Families on Transformation Graphs: $G^{xy+}$ and $G^{xy-}$

Abstract

In this study, transformation graphs obtained from the concept of the total graph and the result of its paired domination number for some special graph families are discussed. If a subset $S$ of the vertex set of the graph $G$ dominates and the induced subgraph $⟨S⟩$ has a perfect matching that covers every vertex of the graph, then $S$ is called a paired-dominating set of $G$. A paired dominating set with the smallest cardinality is denoted by $\gamma_{pr}$-set. Haynes and Slater introduced paired domination parameters. The present study commences with assessing outcomes stemming from eight permutations within the realm of path graphs. Subsequently, building upon this foundational structure, the results are extrapolated from the realm of cycle transformation graph structures based on findings from path transformation graphs.