Some conjectures of Graffiti.pc on total domination

We limit our discussion to graphs that are simple and finite of order . Although 8 we often identify a graph with its set of vertices, in cases where we need to be K explicit we write . A set of vertices of is said to Z ÐKÑ Q K dominate K provided each vertex of is either in or adjacent to a vertex of . K Q Q The domination number of is the minimum order of a dominating set. A K dominating provided each vertex of set of is said to Q K totally dominate K K is adjacent to a vertex of . The of is the minimum Q K total domination number order of a totally dominating set. The total domination number is denoted by # # > > œ ÐKÑ. The minimum order of a dominating set is denoted by connected # # œ ÐKÑ. Other definitions will be introduced immediately prior to their first appearance. The total domination number of a graph was first introduced in [3]. This invariant remains of interest to researchers as evidenced by numerous recent papers. Various upper and lower bounds on total domination have been discovered. The domination number has, of course, been well studied ([15], [16]). Graffiti, a computer program that makes conjectures, was written by S. Fajtlowicz and dates from the mid-1980's. Graffiti.pc was written by E. DeLaViña in 2001. The operation of Graffiti.pc and its similarities to Graffiti are described in [4] and [5]; its conjectures can be found in [6]. A numbered, annotated listing of several hundred of Graffiti's conjectures can be found in