Total Domination Good Vertices in Graphs

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A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. A vertex that is contained in some minimum total dominating set of a graph G is a good vertex, otherwise it is a bad vertex. We determine for which triples (x, y, z) there exists a connected graph G with γt(G) = x and with y good vertices and z bad vertices, and we give graphs realizing these triples.