Proper total domination in graphs

A vertex u in a graph G totally dominates a vertex v if v is adjacent to u.A subset S of the vertex set of a graph G is a total dominating set for G if every vertex of G is totally dominated by at least one vertex of S. The minimum cardinality of a total dominating set for G is the total domination number γt(G) of G.If S is a total dominating set of a graph G, then σS(v) denotes the number of vertices in S that totally dominate v.A total dominating set S in a graph G is called a proper total dominating set if σS(u) = σS(v) for every two adjacent vertices u and v of G.Not all graphs possess a proper total dominating set.Those paths and cycles possessing a proper total dominating set are determined.It is shown that every n × m grid Pn Pm (the Cartesian product of paths Pn and Pm of order n and m respectively) with n ≥ m ≥ 2 has a proper total dominating set.Also, for every r-regular bipartite graph H where r ≥ 2, the graph H P2 has a proper total dominating set.The minimum cardinality of a proper total dominating set in G is the proper total domination number γpt(G).All pairs a, b, of positive integers are determined for which there is a graph G with a proper total dominating set such that γt(G) = a and γpt(G) = b.