Edge-to-vertex detour number of a graph

Abstract

For two vertices u and v in a graph G = (V,E), the detour distance D(u, v)is the length of a longest u–v path in G. A u–v path of length D(u, v) is called a u–v detour. For subsets A and B of V , the detour distance D(A,B)is defined as D(A,B) = min{D(x, y) : x ∈ A, y ∈ B}. A u–v path of length D(A,B) is called an A–B detour joining the sets A, B ⊆ V where u ∈ A and v ∈ B. A vertex x is said to lie on an A–B detour if x is a vertex of an A–B detour. A set S ⊆ E is called an edge-to-vertex detour set if every vertex of G is incident with an edge of S or lies on a detour joining a pair of edges of S. The edge-to-vertex detour number dn_2(G) of G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn_2(G) is an edge-to-vertex detour basis of G. Certain general properties of these concepts are studied. The edge-to-vertex detour numbers of certain classes of graphs are determined. Its relationship with the detour diameter is discussed and it is proved that for each triple D, k, q of integers with 2 ≤ k ≤ q − D + 2 and D ≥ 4 there is a connected graph G of order p with detour diameter D and dn_2(G) = k. It is also proved that for any three positive integers a, b, k with k ≥ 2 and a < b ≤ 2a, there is a connected graph with detour radius a, detour diameter b and dn_2(G) = k.