Volume :8 , Issue :1 ,Page :24-29

Abstract : A detour dominating set ܵ in a connected graph ܩ is called a minimal detour dominating set of ܩ if no proper subset of ܵ is a detour dominating set of ܩ . The upper detour domination number ߛ ௗ ା ሻܩሺ of ܩ is the maximum cardinality of a minimal detour dominating set of ܩ . Some general properties satisfied by this concept are studied. For a connected graph ܩ of order p with upper detour domination number p is characterized. It is shown that for every two positive integers ܽ and ,ܾ with 2 ൑ ܽ ൑ ܾ, there exists a connected graph ܩ with ߛ ௗ ሺܩሻ ൌ ܽ and ߛ ௗ ା ሺܩሻ ൌ ܾ.