Restrained Domination Coalition Number of Paths and Cycles

A restrained domination coalition (or simply rdc) consists of two disjoint subsets of vertices R1 and R2 of a graph Gh. Neither R1 nor R2, on its own, is a restrained dominating set (RD-set). However, when combined, they together form an RD-set for the graph. A restrained domination coalition partition (rdcp) is a vertex partition πr = {R1,R2, ..,Rl} where each element of Ri ∈ πr is either an RD-set consisting of a single vertex, or a non-RD-set that forms an rdc with a set Rj in πr. In this work, we initiated the concept of rdc and rdc-graph. We further proved the existence of rdc for any simple graph. Moreover, we determine the exact value of this parameter in special graph families such as complete multipartite graphs, paths and cycles, while establishing the relation between rdc-number and graph invariants like vertex degree. We further characterized the rdc-graphs of paths. This study applies rdc-partitioning to cybersecurity, structuring networks into collaborative security clusters that detect, contain, and neutralize threats in real time.