Asian Journal of Mathematics and Computer Research

Abstract

In this paper the concept of semi-total monophonic domination number of a graph is introduced. A set of vertices  of a graph  is called a total monophonic set if  is a monophonic set and its induced subgraph has no isolated vertices. The minimum cardinality of all total monophonic sets of  is called the total monophonic number and is denoted by.  A set of vertices  in  is called a monophonic dominating set if  is both a monophonic set and a dominating set. The minimum cardinality of a monophonic dominating set of  is its monophonic domination number and is denoted by . A monophonic dominating set of size  is said to be a  set. A set  of vertices in a graph  with no isolated vertices is said to be a semi-total monophonic set of   if it is a monophonic set of   and every vertex in  is within distance 2 of another vertex of . The semi-total monophonic AMS Subject classification:  05C12 number, denoted by , is the minimum cardinality of a semitotal monophonic dominating set of .