Properties 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. Properties of γtECγtEC graphs are established.