The maximum diameter of total domination edge-critical graphs

A set SS of vertices in a graph GG is a total dominating set of GG if every vertex of GG is adjacent to some vertex in SS. The minimum cardinality of a total dominating set of GG is the total domination number γt(G)γt(G) of GG. The graph GG is total domination edge-critical if for every edge ee in the complement of GG, γt(G+e)<γt(G)γt(G+e)<γt(G). We call such graphs γtECγtEC. If GG is γtECγtEC and γt(G)=kγt(G)=k, we say that GG is ktECktEC. For k≥2k≥2, we show that the maximum diameter of a ktECktEC graph is at least ⌊3(k−1)/2⌋⌊3(k−1)/2⌋ and this bound is sharp for small k≤6k≤6.