On the diameter of dot-critical graphs
Doost Ali Mojdeh, Somayeh Mirzamani · Opuscula Mathematica · 2009
A graph G is k-dot-critical (totaly k-dot-critical) if G is dot-critical (totaly dot-critical) and the domination number is k.In the paper [T.Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306(2006), 11-18] the following question is posed: What are the best bounds for the diameter of a k-dot-critical graph and a totally k-dot-critical graph G with no critical vertices for k ≥ 4? We find the best bound for the diameter of a k-dot-critical graph, where k ∈ {4, 5, 6} and we give a family of k-dot-critical graphs (with no critical vertices) with sharp diameter 2k -3 for even k ≥ 4.