DefinePK

Abstract

For two vertices u and v 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 a detour set of if every vertex in lies on a detour joining a pair of vertices of . The detour number of G is the minimum order of its detour sets and any detour set of order is a detour basis of . A set is called a connected detour set of if S is detour set of and the subgraph induced by S is connected. The connected detour number of is the minimum order of its connected detour sets and any connected detour set of order is called a connected detour basis of . Graphs G with detour diameter are characterized when , , or . A subset of a connected detour basis of is a forcing subset for if is the unique connected detour basis containing . The forcing connected detour number of is the minimum cardinality of a forcing subset for . The forcing connected detour number of is , where the minimum is taken over all connected detour bases in . The forcing connected detour numbers of certain classes of graphs are determined. It is also shown that for each pair , of integers with and , there is a connected graph with and .