Edge-to-vertex m-detour monophonic number of a graph.

Abstract

For a connected graph G = (V, E) of order at least three, the monophonic distance d_m(u, v) is the length of a longest u − v monophonic path in G. A u − v path of length d_m(u, v) is called a u − v detour monophonic. For subsets A and B of V, the m-monophonic distance D_m(A, B) is defined as D_m(A, B) = max{d_m(x, y) : x ∈ A, y ∈ B}. A u − v path of length D_m(A, B) is called a A − B m-detour monophonic path joining the sets A, B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called an edge-to-vertex m-detour monophonic set of G if every vertex of G is incident with an edge of S or lies on a m-detour monophonic path joining a pair of edges of S. The edge-to-vertex mdetour monophonic number Dm_ev(G) of G is the minimum order of its edge-to-vertex m-detour monophonic sets and any edge-to-vertex m-detour monophonic set of order Dm_ev(G) is an edge-to-vertex mdetour monophonic basis of G. Some general properties satisfied by this parameter are studied. The edge-to-vertex m-detour monophonic number of certain classes of graphs are determined. It is shown that for positive integers r, d and k ≥ 4 with r < d, there exists a connected graph G such that rad_m(G) = r, diam_m(G) = d and Dm_ev(G) = k