EDGE-TO-VERTEX DETOUR MONOPHONIC NUMBER OF A GRAPH

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. For subsets A and B of V , the monophonic distance d_m(A, B) is defined as d_m(A, B) = min{d_m(x, y) :x ∈ A, y ∈ B}. A u − v path of length d_m(A, B) is called an A − B 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 detour monophonic set of G if every vertex of G is incident with an edge of S or lies on a detour monophonic joining a pair of edges of S. The edge-to-vertex detour monophonic number dm_{ev}(G) of G is the minimum order of its edge- to-vertex detour monophonic sets and any edge-to-vertex detour monophonic set of order dm_{ev}(G) is an edge-to-vertex detour monophonic basis of G. Certain general properties of these concepts are studied. It is shown that for each pair of integers k and q with 2 ≤ k ≤ q,there exists a connected graph G of order q + 1 and size q with dm_{ev}(G) = k.