PARTITIONING A GRAPH INTO MONOPOLY SETS

AHMED MOHAMMED NAJI, SONER NANDAPPA D

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Abstract

In a graph G = (V, E), a subset M of 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  which is different from {1, 2, 4}, there exists a graph G of order n = k having μ(G) = 3.

Keywords

vertex degrees, distance in graphs, graph polynomials.