THE UPPER RESTRAINED DETOUR ONOPHONICDOMINATION NUMBER OF A GRAPH

Abstract

In this paper the concept of minimal restrained detour monophonic domination number ????of a graph ???? is introduced. For a connected graph ???? = (????, ????) of order at least two, a minimal 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 minimal restrained detour monophonic dominating set of ???? is the minimal 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 three positive integers ????, ????, ???? and ????, with 2 ≤ ???? ≤ ???? ≤ ???? ≤ ????, there is a connected graph ???? with ????????(????) = ????, ????????????(????) = ????,????????????????(????) = ???? and ???????????????? +(????) = ????.