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 f(edm)(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 f(edm)(G) = min{f(edm)(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 f(edm)(G) = a and edm(G) = b.