Weak and strong domination on some graphs

Abstract

Let G = (V(G), E(G)) be a graph and uvεE. A subset D ⊆ V of vertices is a dominating set if every vertex in V − D is adjacent to at least one vertex of D. The domination number is the minimum cardinality of a dominating set. Let u and v be elements of V. Then, u strongly dominates u and v weakly dominates u if (i)uvεE and (ii)deg(u) ≥ deg(v). A set D ⊆ V is a strong (weak) dominating set (sd-set)(wd-set) of G if every vertex in V − D is strongly dominated by at least one vertex in D. The strong (weak) domination number γs(γw) of G is the minimum cardinality of a sd-set (wd-set). In this paper, the strong and weak domination numbers of comet, double comet, double star and theta graphs are given. The theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, MST construction and real-time animation.