Equitable Power Edge Domination Number of Duplicate and Extended Duplicate Graph of Some Special Graphs

Abstract

The duplicate graph, denoted as, is formed by duplicating each vertex in the original graph. [2] E. Sampath Kumar presented the idea of a duplicate graph and demonstrated several outcomes using it. A duplicate graph is a graph that is created by duplicating the vertices of an existing graph while preserving the connections connecting them. With  denoting a collection of vertices and denoting a set of edges, we refer to  as a graph.  [7] An Equitable Power Dominating set S in graph  is a subset of vertices in  is a power dominating set if, for every vertex  in  and each vertex  in  that is adjacent to , the absolute difference between the degrees of  and  is less than or equal to 1, expressed as . Based on this definition, a new concept called "Equitable Power Edge Domination" has been introduced. It's defined as follows: An edge that is not in the set  is considered an "Equitable Power Edge Dominating set" if it's observed to be adjacent to an edge  such that   The smallest number of elements in a collection of graphs exhibiting equitable power edge domination is denoted as the equitable power edge domination number of the graph, represented by (????). In this study, the Equitable Power Edge Domination number (EPEDN) of certain duplicate graphs is determined and represented as . Extended duplicate graph (EDG) is introduced by Vijayakumar et.al and proved the existence of harmonious labelling of star graph. In simpler terms, to determine the edges in the duplicate graph  you look at the original graph  and consider pairs of vertices  connected by edges in . If both the pairs and  are edges in , then the edge  in G. The Equitable Power Edge Domination Number (EPEDN) for several types of duplicate graphs, including a path, ladder, wheel, twig, comb, and star, has been defined. A comprehensive study of special graphs has also been done, along with a comparison of EPED and EPED of duplicate graphs. EPED has a lower minimal cardinality than EPED of duplicate graph, according to the findings. We are using this extended strategy for obtaining the EPED number of Extended duplicate graphs. The Extended Duplicate Graph (EDG) is a graph with duplicated nodes, maintained edges due to the bijective function, and an extra edge between nodes  and . An Extended Duplicate Graph (EDG) is created as a consequence of this procedure, which extends the graph beyond its original form while maintaining the structure of the original graph (with duplicated nodes and intact edges).