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 p of edge-disjoint subgraphs G(1),G(2),...,G(n) of G such that every edge of G belongs to exactly one G(i),(1 <= i <= n). The decomposition pi = {G(1),G(2),...,G(n)} of a connected graph G is said to be a distinct edge geodetic decomposition if g(1)(Gi) not equal g(1)(G(j)),(1 <= i not equal j <= n). The maximum cardinality of pi is called the distinct edge geodetic decomposition number of G and is denoted by pi(dg1)(G), where g(1)(G) is the edge geodetic number of G. Some general properties satisfied by this concept are studied. Connected graphs of pi(dg1)(G) >= 2 are characterized and connected graphs of order pi with pi(dg1)(G) = p -2 are characterized.