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On the diameter of dot-critical graphs

Mojdeh D. A., Mirzamani S.

Opuscula Mathematica

2009

Vol. 29, no. 2

165-175

A graph G is k-dot-critical (totally k-dot-critical) if G is dot-critical (totally 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.