THE UPPER EDGE TO EDGE DETOUR DOMINATION NUMBER OF A GRAPH

Abstract

An edge-to-edge detour dominating set ???? in a connected graph ???? is called a minimal edge-to-edge detour dominating set if no proper subset of ???? is an edge-to-edge detour dominating set of ????. The upper edge-to-edge detour domination number ???????????????? + (????) of ???? is the maximum cardinality of a minimal edge-to-edge detour dominating set of ????. Some general properties satisfied by this concept are studied. The upper edge-to-edge detour domination number of some standard graphs are determined. It is shown that for any two positive integers ???? and ???? with 2 ≤ ???? ≤ ????, there exists a connected graph ???? such that ???????????????? (????) = ???? and ???????????????? + (????) = ????.