Connected Weak Edge Detour Number of a Graph

Abstract

Certain general properties of the detour distance, weak edge detour set, connected weak edge detour set, connected weak edge detour number and connected weak edge detour basis of graphs are studied in this paper. Their relationship with the detour diameter is discussed. It is proved that for each pair of integers k and n with 2 <= k <= n, there exists a connected graph G of order n with cdn_w(G)=k. It is also proved that for any three positive integers R,D,k such that k >= D and R < D <= 2R, there exists a connected graph G with rad_D (G) = R, diam_D G = D and cdn_w(G)=k.