THE CONNECTED RESTRAINED DETOUR MONOPHONIC DOMINATION NUMBER OF A GRAPH

Abstract

In this paper the concept of connected restrained detour monophonic domination number ????of a graph ???? is introduced. For a connected graph ???? = (????, ????) of order at least two, a connected restrained detour monophonic dominating set ???? of a graph???? is a detour monophonic dominating set such that either ???? = ???? or the sub graph induced by ???? – ???? has no isolated vertices. The minimum cardinality a connected restrained detour monophonic dominating set of ???? is the connected restrained detour monophonic domination number of ???? and is denoted by ???????????????????? (????). We determine bounds for it and characterize graphs which realize these bounds. It is shown that For any positive integers ????, ???? and ???? ≥ 6 with ???? < ????, there exists a connected graph ????with ????????????????(????) = ????, ????????????????????(????) = ???? and ???????????????????? (????) = ????.