Gallai-type theorems in domination and strong domination parameters

Let G = (V, E) be a graph and u, v ∈ V. If uv ∈ E, we say that u and v dominate each other. If uv ∈ E and deg u ≥ deg v, we say that u strongly dominates v. A subset D of V is called a dominating set of G if every vertex v ∈ V - D is dominated by some u ∈ D. A subset D of V is called a strong dominating set of G if every vertex v ∈ V - D is strongly dominated by some u ∈ D. The domination number γ(G) of G is the cardinality of its smallest dominating set. Similarly the strong domination number γs(G) of G is denned. Gallai-type theorems are of the form x(G) + y(G) = n, where x(G),y(G) are parameters denned on a graph G with n vertices. Let δ(G) and Δ(G) respectively denote the minimum and maximum degree of vertices of a graph G. In this paper, we give Gallai-type theorems in the form γ(G) + Δ(G) = n; γs (G) + Δ(G) = n.