On edge detour graphs

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. A set S ⊆ V is called an edge detour set if every edge in G lies on a detour joining a pair of vertices of S. The edge detour number dn1(G) of G is the minimum order of its edge detour sets and any edge detour set of order dn1(G) is an edge detour basis of G. A connected graph G is called an edge detour graph if it has an edge detour set. It is proved that for any non-trivial tree T of order p and detour diameter D, dn1(T ) ≤ p−D +1 and dn1(T ) = p−D +1 if and only if T is a caterpillar. We show that for each triple D, k, p of integers with 3 ≤ k ≤ p − D + 1 and D ≥ 4, there is an edge detour graph G of order p with detour diameter D and dn1(G) = k. We also show that for any three positive integers R, D, k with k ≥ 3 and R < D ≤ 2R, there is an edge detour graph G with detour radius R, detour diameter D and dn1(G) = k. Edge detour graphs G with detour diameter D ≤ 4 are characterized when dn1(G) = p − 2 or dn1(G) = p − 1.