On the edge monophonic number of a graph

For a connected graph G = (V, E), an edge monophonic set of G is a set M? V(G) such that every edge of G is contained in a monophonic path joining some pair of vertices in M. The edge monophonic number m1 (G) of G is the minimum order of its edge monophonic sets and any edge monophonic set of order m1 (G) is a minimum edge monophonic set of G. Connected graphs of order p with edge monophonic number p are characterized. Necessary condition for edge monophonic number to be p ? 1 is given. It is shown that for every two integers a and b such that 2 ? a ? b, there exists a connected graph G with m(G) = a and m1 (G) = b, where m(G) is the monophonic number of G.