DefinePK

Abstract

For two vertices and in a graph , the detour distance is the length of a longest path in . A path of length is called a detour. A set is called an edge detour set if every edge in lies on a detour joining a pair of vertices of . The edge detour number of G is the minimum order of its edge detour sets and any edge detour set of order is an edge detour basis of . A connected graph is called an edge detour graph if it has an edge detour set. A subset of an edge detour basis of an edge detour graph is called a forcing subset for if is the unique edge detour basis containing . A forcing subset for of minimum cardinality is a minimum forcing subset of . The forcing edge detour number of , is the minimum cardinality of a forcing subset for . The forcing edge detour number of , is , where the minimum is taken over all edge detour bases in . The general properties satisfied by these forcing subsets are discussed and the forcing edge detour numbers of certain classes of standard edge detour graphs are determined. The parameters and satisfy the relation and it is proved that for each pair , of integers with and , there is an edge detour graph with and .