Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

Abstract

Let G=(V, E) be a graph. A dominating set S of graph G is defined as a set of vertices such that every vertex in V\S is adjacent to at least one vertex in S. The domination number of graph G, denoted as γ(G), corresponds to the size of the smallest dominating set within G. In other words, γ(G) represents the number of vertices required in the minimum dominating set to cover all other vertices in the graph G. In the graph G, our objective is to position a protector at each vertex within a subset S of V, ensuring that S forms a dominating set, effectively covering all other vertices in G. Moreover, in the event that a protector positioned at vertex 𝑣 needs to move along an edge to protect an unguarded vertex u, the arrangement of protectors should maintain the property of forming a dominating set for the graph. In other words, the movement of protectors should maintain the property of domination within the graph, ensuring efficient coverage and defense across the network. The bare minimum of security guards is necessary to protect all vertices in the graphs. In this article, we find the bounds for domination, independent domination number (IDN), connected domination number(CDN), total domination number(TDN), and the secure domination number(SDN) denoted byγ(A n ), γ i (A n ), γ c (A n ), γ t and γ s (A n ) respectively for the antiprism graph, where A n denoted the 4 -regular graph with girth 3. We further establish that the TDN is greater than or equal to the SDN of the antiprism graph for 𝑛 ≥ 3.