forcing edge detour monophonic number of a graph

For a connected graph g = (v, e) of order at least two, an edge detour monophonic set of g is a set s of vertices such that every edge of g lies on a detour monophonic path joining some pair of vertices in s. the edge detour monophonic number of g is the minimum cardinality of its edge detour monophonic sets and is denoted by edm(g). a subset t of s is a forcing edge detour monophonic subset for s if s is the unique edge detour monophonic set of size edm(g) containing t. a forcing edge detour monophonic subset for s of minimum cardinality is a minimum forcing edge detour monophonic subset of s. the forcing edge detour monophonic number fedm(s) in g is the cardinality of a minimum forcing edge detour monophonic subset of s. the forcing edge detour monophonic number of g is fedm(g) = min{fedm(s)}, where the minimum is taken over all edge detour monophonic sets s of size edm(g) in g. we determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs. it is shown that for every pair a, b of positive integers with 0 ≤ a < b and b ≥ 2, there exists a connected graph g such that fedm(g) = a and edm(g) = b.