Examining the number of super domination in graphs

For a given simple graph G = (V, E), a dominating set is defined as a subset D ⊆ V such that every vertex in V \ D is adjacent to at least one vertex in D. The dominating set problem seeks to identify a dominating set of minimum cardinality, where this minimum size is referred to as the domination number of the graph. A dominating set D is called a super-dominating set if, for every vertex u ∈ V \ D, there exists a vertex v ∈ D such that N (v) ∩ (V D) = u. The super domination number of a graph G, denoted γsp(G), is defined as the minimum cardinality of a super dominating set. The middle graph of a graph G, denoted M (G), is constructed by introducing a new vertex corresponding to each edge of G and connecting these new vertices by edges if their corresponding edges in G are adjacent. This study aims to thoroughly investigate the super domination number of middle graphs, generalized tranformation grahs and generalized petersen graphs. By considering the structural properties of these graph classes, general theoretical bounds for the super-domination number have been established. These limits are expressed in terms of the vertex count of the graph, the edge count and other structural parameters, offering new insights into the super-dominance behavior of the middle graphs. Furthermore, the super-domination number has been analyzed for the middle graphs of specific graph families, including path graphs (Pn), cycle graphs (Cn), complete graphs (Kn), and star graphs (K1,n). The results for these graph families are presented either as exact values, shedding light on the behavior of the super domination number in these graph classes. The study explores the relationship between the properties of a simple graph G and the super-domination number of its middle graph, generalized transformation graph and generalized petersen graph. The findings are intended to contribute to a deeper understanding of the super-domination number of middle graphs in both theoretical and practical contexts.