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