Connected Dom-\(k\)-forcing Sets in Graphs

A vertex subset Dk f in a graph G is referred to as a dom-k-forcing set if it satisfies the properties of both a k-forcing set and a dominating set. The dom-k-forcing number of a graph G, denoted as Fdk(G), represents the smallest possible size of a dom-k-forcing set. A connected dom-k-forcing set of a graph G, is a dom-k-forcing set of G that induces a sub graph of G which is connected. The connected dom-k-forcing number of G, Fcdk(G), is the minimum size of a connected dom-k-forcing set. In this paper, we introduce the study of connected dom-k-forcing sets in graphs. We derived formulas for finding precise values of connected dom-k- forcing number of tree, grid graph, prism graph, gear graph, path, cycle, etc. Additionally, we characterize connected dom-k-forcing sets in certain special graphs, deriving formulas for connected dom-k-forcing number based on these characterizations.