The Forcing Weak Edge Detour Number of a Graph

Abstract

For two vertices u and v in a graph G = (V,E), the distance d(u,v) and detour distance D(u,v) are the length of a shortest or longest u − v path in G, respectively, and the Smarandache distance d i(u,v) is the length d(u,v)+ i(u,v) of a u −v path in G, where 0 ≤ i(u,v) ≤ D(u,v) − d(u,v). A u − v path of length d i(u,v), if it exists, is called a Smarandachely u − v i-detour. A set S ⊆ V is called a Smarandachely i-detour set if every edge in G has both its ends in S or it lies on a Smarandachely i-detour joining a pair of vertices in S. In particular, if i(u,v) = 0, then d i(u,v) = d(u,v); and if i(u,v) = D(u,v) − d(u,v), then d i(u,v) = D(u,v). For i(u,v) = D(u,v) − d(u,v), such a Smarandachely i-detour set is called a weak edge detour set in G. The weak edge detour number dnw(G) of G is the minimum order of its weak edge detour sets and any weak edge detour set of order dnw(G) is a weak edge detour basis of G. For any weak edge detour basis S of G, a subset T ⊆ S is called a forcing subset for S if S is the unique weak edge detour basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing weak edge detour number of S, denoted by fdnw(S), is the cardinality of a minimum forcing subset for S. The forcing weak edge detour number of G, denoted by fdnw(G), is fdnw(G) = min{fdnw(S)}, where the minimum is taken over all weak edge detour bases S in G. The forcing weak edge detour numbers of certain classes of graphs are determined. It is proved that for each pair a,b of integers with 0 ≤ a ≤ b and b ≥ 2, there is a connected graph G with fdnw(G) = a and dnw(G) = b.