distinct edge geodetic decomposition in graphs

Let g=(v,e) be a simple connected graph of order p and size q. a decomposition of a graph g is a collection π= g1,g2,…,gn of g such that every edge of g belongs to exactly one gi,(1≤i≤n). the decomposition π={g1,g2,…,gn} of a connected graph g is said to be a distinct edge geodetic decomposition if g1(gi)≠g1(gj),(1≤i≠j≤n). the maximum cardinality of π is called the distinct edge geodetic decomposition number of g and is denoted by πdg1(g), where g1(g) is the edge geodetic number of g. some general properties satisfied by this concept are studied. connected graphs of πdg1(g)≥2 are characterized and connected graphs of order p with πdg1(g)=p−2 are characterized.