On Restrained Coalitions in Graphs: Bounds and Exact Values

A subset D V is a dominating set of a graph G with vertex set V if every vertex v V \ D is adjacent to a vertex in D. Two subsets of V form a coalition if neither of them is a dominating set, but their union is a dominating set.A coalition partition of G is its vertex partition such that every non-dominating set of is a member of some coalition, and every dominating set is a single-vertex set in .The coalition number C(G) of a graph G is the maximum cardinality of its coalition partitions.A subset R V is a restrained dominating set if R is a dominating set and any vertex of V \ R has at least one neighbor in V \ R. Restrained dominating coalition, restrained dominating partition and restrained coalition number RC(G) are defined by the same way.In this paper, we prove that RC(G) C(G) for an arbitrary graph G.In addition, some previous results from [A. H. S. Nesam, S. Amutha, N. Anbazhagan, Stat.Optim.Inf.Comput.14 (2025) 3409-3417] are corrected by determining the exact value of the restrained coalition number of cycles.