THE RESTRAINED MONOPHONIC NUMBER OF A GRAPH

A set S of vertices of a connected graph G is a monophonic set of G if each vertex v of G lies on a x - y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). A restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V - S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by [m.sub.r](G). We determine bounds for it and determine the same for some special classes of graphs. Further, several interesting results and realization theorems are proved. Keywords: monophonic set, monophonic number, restrained monophonic set, restrained monophonic number. AMS Subject Classification: 05C12.