The diameter of total domination vertex critical graphs (Q1887647)

A graph \(G\) with total domination number \(\gamma_t(G)=k\) is called \(k\)-\(\gamma_t\)-critical if \(\gamma_t(G-v)<k\) for all vertices \(v\) for which \(G-v\) contains no isolated vertex. The authors characterize connected \(k\)-\(\gamma_t\)-critical graphs having at least one endvertex and derive a sharp upper bound on their diameter. Furthermore, they prove that the maximum possible diameter of connected \(k\)-\(\gamma_t\)-critical graphs in general lies between \(5k/3-O(1)\) and \(2k-3\) and determine the exact values for \(k\leq 8\).