On 4-γtγt-critical graphs of diameter two

A vertex subset SS of graph GG is a total dominating set   of GG if every vertex of GG is adjacent to a vertex in SS. For a graph GG with no isolated vertex, the total domination number   of GG, denoted by γt(G)γt(G), is the minimum cardinality of a total dominating set. A total dominating set of cardinality γt(G)γt(G) is called a γt(G)-set. A graph GG with no isolated vertex is total domination vertex critical   if for any vertex vv of GG that is not adjacent to a vertex of degree one, the total domination number of G−vG−v is less than the total domination number of GG. We call these graphs γt-critical. If such a graph GG has total domination number kk, then we call it k-γt-critical. In this note we study 4-γt-critical connected graphs GG of diameter two. We prove that such graphs with minimum at least two have order at least 10, and we characterize all 44-γtγt-criticalcritical connected graphs of order 10 with maximum degree 55. Moreover, we obtain some 4-γt-critical connected graphs of order 10 with maximum degree 44 and for any integer k≥2k≥2, n=3k+5n=3k+5, there exists a 4-γt-critical graph GG of order nn with diam(G)=2(G)=2.