Partitioning a graph into monopoly sets

Abstract

In a graph G = (V, E), a set M ⊆ V (G) is said to be a monopoly set of G if every vertex v ∈ V − M has, at least, d(v)/2 neighbors in M. The monopoly size of G, denoted by mo(G), is the minimum cardinality of a monopoly set. In this paper, we study the problem of partitioning V (G) into monopoly sets. An M-partition of a graph G is the partition of V (G) into k disjoint monopoly sets. The monatic number of G, denoted by µ(G), is the maximum number of sets in M-partition of G. It is shown that 2 ≤ µ(G) ≤ 3 for every graph G without isolated vertices. The properties of each monopoly partite set of G are presented. Moreover, the properties of all graphs G having µ(G) = 3, are presented. It is shown that every graph G having µ(G) = 3 is Eulerian and have χ(G) ≤ 3. Finally, it is shown that for every integer k /∈ {1, 2, 4}, there exists a graph G of order n = k having µ(G) = 3.