Super Coalition Number in Graphs

Abstract

We introduce and investigate the structural properties of super coalition partitions in graphs, a novel direction that bridges cooperative resource deployment with rigid domination criteria. Based on the foundational concept of super domination, a super coalition partition is defined as a vertex set partitioning $Υ= \{A_1, A_2, \ldots, A_k\}$ such that no single class $A_i$ constitutes a valid super dominating set, yet every class can be paired with at least one distinct partner class $A_j$ to form a union $A_i \cup A_j$ that achieves full super domination over the graph. The super coalition number, denoted by $C_s(G)$, represents the maximum possible cardinality of such a partition. In this paper, we establish general operational bounds for $C_s(G)$ using the underlying order and the super domination number $γ_{sp}(G)$, demonstrate its relation to the super domatic number $d_{sp}(G)$, analyze its computational complexity proving its NP-complete nature under general conditions, and provide exact determinations for key standard graph architectures including paths, cycles, complete graphs, stars, wheels, and friendship configurations. We conclude by proving that the super coalition number can grow arbitrarily large.