The Forcing Edge-to-vertex Detour Number of a Graph

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-tovertex 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. A subset T of an edge-to-vertex detour basis S is called a forcing subset for S if S is the minimum forcing subset of S. The forcing edge-to-vertex detour number of S, denoted by fdn 2 (S), is the cardinality of a minimum forcing subset for S. The forcing edge-to-vertex detour number of G, denoted by fdn 2 (G), is fdn 2 (G)= min {fdn 2 (S)}, where the minimum is taken over all edge-to-vertex detour bases S in G. The forcing edge-to-vertex detour numbers of certain standard graphs are obtained. It is shown that for every pair a, b of integers with 0 ≤ a ≤ b and b ≥ 2 there exists a connected graph G with fdn 2 (G) = a and dn 2 (G) = b. AMS SUBJECT CLASSIFICATION: 05C12.