THE TOTAL MONOPHONIC DOMINATION NUMBER OF A GRAPH

In this paper the concept of total monophonic domination number of a graph is introduced. A set of vertices M of a graph G is called a total monophonic set if M is a monophonic set and its induced subgraph has no isolated vertices. The minimum cardinality of all total monophonic sets of M is called the total monophonic number and is denoted by mt (G). A total monophonic dominating set is a monophonic dominating set and its induced subgraph has no isolated vertices. The minimum cardinality of all such total monophonic domination sets of G is called the total monophonic domination number and is denoted by γmt (G). It is shown that for any positive integers 2<a<b<c, and a+b>c, there exists a connected graph G such that m(G)=a, γm (G)=b and γmt (G)=c. Also, for every pair k,p of integers with 3≤k≤p, there exists a connected graph G of order p such that γmt (G)=k.