Connected domination critical graphs with a block having maximum number of cut vertices

A set D of vertices in a graph G is a dominating set if every vertex in V(G)−D is adjacent to a vertex in D. If the subgraph induced by the set D is connected, then D is a connected dominating set in G. The connected domination number of G, γ_c(G), is the minimum cardinality of a connected dominating set of G. A graph G is k-γ_c-critical if γ_c(G)=k and γ_c(G+uv)<k for every pair of non-adjacent vertices u and v of G. Let G be a k-γ_c-critical graph, ζ the number of cut-vertices of G, and ζ₀ the maximum number of cut-vertices can be contained in one block (a maximal 2-connected subgraph) of G. It is known that [Formula presented]. Let A be the class of k-γ_c-critical graphs containing a block B₀ which has [Formula presented] cut-vertices. Further, for integer 0≤i≤2, we let A_i be the subclass of A such that k≡i(mod3). In this paper, we show that ζ≤{ζ₀+2ifG∈A₀,ζ₀ifG∈A₁,ζ₀+1ifG∈A₂.Further, we characterize all graphs in the classes A_i whose number of cut-vertices achieve the upper bound for each i∈{0,1,2}.