Realizability in Proper Total Domination in Graphs

A vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A set S of vertices of G is a total dominating set if every vertex of G is totally dominated by some vertex of S. The minimum cardinality of a total dominating set of G is its total domination number t(G) of G.A proper total dominating set (or pt-dominating set) of G is a total dominating set S where no two adjacent vertices of G are totally dominated by the same number of vertices of S. The minimum cardinality of a pt-dominating set of G is the pt-domination number pt(G), while the maximum cardinality of such a set is the upper pt-domination number pt(G).While every graph without isolated vertices has a total dominating set, this need not be the case for pt-dominating sets.For every graph G having a pt-dominating set, t(G) pt(G) pt(G).It is shown that for every pair a, b of integers with 3 a b, there exists a connected graph G with pt(G) = a and pt(G) = b.Furthermore, those triples a, b, c of integers with 2 a b c are studied for which there is a connected graph G with t(G) = a, pt(G) = b, and pt(G) = c.