THE CONNECTED RESTRAINED DETOUR EDGE MONOPHONIC DOMINATION NUMBER OF A GRAPH

Abstract

In this paper the concept of connected restrained detour edge monophonic domination number ????of a graph ???? is introduced. For a connected graph ???? = (????, ????) of order at least two, a connected restrained detour edge monophonic dominating set ???? of a graph ???? is a detour edge monophonic dominating set such that either ???? = ???? or the sub graph induced by ???? – ???? has no isolated vertices. The minimum cardinality a minimal restrained detour edge monophonic dominating set of ???? is the minimal restrained detour edge monophonic domination number of ???? and is denoted by ???????????????????????? (????). We determine bounds for it and characterize graphs which realize these bounds. It is shown thatIf ????, ????, ????, ???? and ???? are positive integers such that 3 ≤ ???? ≤ ???? ≤ ???? ≤ ???? ≤ ???? − 2, then there exists a connected graph ???? of order ????,????????(????) = ????, ????????????(????) = ????, ???????????????? (????) = ???? and ???????????????????? (????) = ????.