Monopolized Domination in Graphs: Structural Foundations, Decompositions, and Equivalence Results

Abstract

We introduce the notion of a monopolized dominating set (MDS) $S$, a neighborhood-based variant of domination in graphs in which every vertex $v \in V(G) \setminus S$ is adjacent to a monopolizer $u \in S$ whose neighborhood is disjoint from that of $v$ ($N(u) \cap N(v) = \varnothing$). We establish a general functional partition theorem for MDS and prove that in triangle-free graphs, every classical dominating set is an MDS, yielding exact formulas for trees, paths, cycles, and complete bipartite graphs. Furthermore, we provide exact structural characterizations linking MDS to several classical concepts: we determine necessary and sufficient conditions under which an open locating-dominating set forms an MDS, characterize when a set simultaneously functions as a vertex cover and an MDS, and prove that the set of centers of a central star decomposition necessarily forms an MDS. These results yield a unified framework connecting domination, vertex covers, and star decompositions.