On the total monophonic number of a graph

Let G = (V, E) be a connected graph of order n: A path P in G which does not have a chord is called a monophonic path. A subset S of V is called a monophonic set if every vertex v in V lies in a x-y monophonic path where x; y is an element of S. If further the induced subgraph G[S] has no isolated vertices, then S is called a total monophonic set. The total monophonic number m(t)(G) and the upper total monophonic number m(t)(+) (G) are respectively the minimum cardinality of a total monophonic set and the maximum cardinality of a minimal total monophonic set. In this paper we determine the value of these parameters for some classes of graphs and establish bounds for the same. We also prove the existence of graphs with prescribed values for m(t)(G) and m(t)(+) (G):