THE SEMI-TOTAL MONOPHONIC DOMINATION NUMBER OF A GRAPH

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 .