Domination in Semigraphs

Let G = (V, X) be a semigraph. A set D ⊆ V is called adjacent dominating set (ad-set) if for every υ ∈ V - D there exists a u ∈ D such that u is adjacent to v. The adjacency domination number γ_a = γ_a(G) is the minimum cardinality of an ad-set of G. Let V_e be the set of all end vertices in G. A set D ⊆ V_e is called end vertex adjacent dominating set (ead-set) if (i) D is an ad-set and (ii) Every end vertex υ ∈ V - D is e-adjacent to some end vertex u ∈ D in G. The end vertex adjacency domination number γ_ea = γ_ea(G) is the minimum cardinality of an ead-set of G. A set D C V is called consecutive adjacent dominating set (cad-set) if for every υ ∈ V - D there exists a u ∈ D such that u is ca-adjacent to v in G. The consecutive adjacency domination number γ_ca = γ_ca(G) is the minimum cardinality of a cad set of G. The above domination parameters are determined for various semigraphs and a few bounds are obtained.