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Applicable Analysis and Discrete Mathematics

Let G = (V,E) be a graph with vertex set V of order n and edge set E. A set S ⊆ V is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and a dominating set of minimum cardinality is called a γ(G)-set. Note that removing a vertex can increase the domination number by more than one, but can decrease it by at most one. We define a graph G to be (γ, k)-critical, if γ(G − S) < γ(G) for any set S of k vertices [1]. Obviously, a (γ, k)-critical graph G has γ(G) ≥ 2, unless in trivial case k = |V (G)| that γ = 1 (for more, we refer to [1–5]). The open neighborhood of a vertex v ∈ V is N(v) = {x ∈ V |vx ∈ E} while N [v] = N(v)∪{v} is the closed neighborhood. So a set S ⊆ V is a dominating set, if V = ⋃ s∈S N [s]. The connectivity of G, written κ(G), is the minimum size of a vertex set S such that G − S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k. A graph is k-edge-connected if every disconnecting set has at least k edges. The edge connectivity of G, written λ(G), is the minimum size of a disconnecting set. We denote the distance between two vertices x and y in G by dG(x, y) and the minimum degree of G with δ(G), the pendant vertex is a vertex of degree 1 and the support vertex is a vertex adjacent