The Forcing Edge Fixed Monophonic Number of a Graph

For an edge xy in a connected graph G of order p ≥ 3, a set S ⊆ V (G) is an xy-monophonic set of G if each vertex v ∈ V (G) lies on either an x-u monophonic path or an y-u monophonic path for some element u in S. The minimum cardinality of an xy-monophonic set of G is the xy-monophonic number of G and is denoted by m_xy(G). A subset T of a minimum xy-monophonic set S of G is an xy-forcing subset for S if S is the unique minimum xy-monophonic set containing T. An xy-forcing subset for S of minimum cardinality is a minimum xy-forcing subset of S. The forcing xy-monophonic number of S, denoted by f_mxy(s), is the cardinality of a minimum xy-forcing subset for S. The forcing xy-monophonic number of G is f_mxy(G) ═ min{f_mxy(s)}, where the minimum is taken over all minimum xy-monophonic sets S in G. We determine bounds for it and find the forcing edge monophonic number for some special classes of graphs. It is shown that for any three positive integer a, b and c with 2 ≤ a ≤ b < c, there exists a connected graph G with f_mxy(G) ═ a, m_xy(G) ═ b and cm_xy(G) ═ c for some edge xy in G, where c_mxy(G) is the connected xy-monophonic number of G.