Domination parameters of a graph with added vertex

Let G = (V, E) be a graph. A subset D ⊆ V is a total dominating set of G if for every vertex y ∈ V there is a vertex x ∈ D with xy ∈ E. A subset D ⊆ V is a strong dominating set of G if for every vertex y ∈ V - D there is a vertex x ∈ D with xy &isin E and degG(x) ≥ degG(y). The total domination number γt(G) (the strong domination number γS(G)) is defined as the minimum cardinality of a total dominating set (a strong dominating set) of G. The concept of total domination was first defined by Cockayne, Dawes and Hedetniemi in 1980 [1], while the strong domination was introduced by Sampathkumar and Pushpa Latha in 1996 [3]. By a subdivision of an edge uv ∈ E we mean removing edge uv, adding a new vertex x, and adding edges ux and vx. A graph obtained from G by subdivision an edge uv ∈ E is denoted by G ⊕ uxvx. The behaviour of the total domination number and the strong domination number of a graph G ⊕ uxvx is developed.