Weak and strong domination on some graphs

Let G = (V(G), E(G)) be a graph and uv epsilon E. A subset D subset of 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)uv epsilon E and (ii)deg(u) >= deg(v). A set D subset of 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 gamma(s)(gamma(w)) 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.