Edge-to-vertex Detour Monophonic Number of a Graph

For a connected graph G = (V,E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u− v monophonic path in G. For subsets A and B of V , the monophonic distance dm(A,B) is defined as dm(A,B) = min{dm(x, y) : x ∈ A, y ∈ B}. A u− v path of length dm(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 dmev(G) of G is the minimum order of its edgeto-vertex detour monophonic sets and any edge-to-vertex detour monophonic set of order dmev(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 dmev(G) = k. Mathematics Subject Classification (2010): 05C12