THE RESTRAINED MONOPHONIC NUMBER OF A GRAPH

THE RESTRAINED MONOPHONIC NUMBER OF A GRAPH

A. P. SANTHAKUMARAN, P. TITUS, K. GANESAMOORTHY

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Abstract. 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 cardi- nality 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_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.