On the diameter of dot-critical graphs

Abstract

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 \(k \geq 4\)? We find the best bound for the diameter of a \(k\)-dot-critical graph, where \(k \in\{4,5,6\}\) and we give a family of \(k\)-dot-critical graphs (with no critical vertices) with sharp diameter \(2k-3\) for even \(k \geq 4\).