Detour homometric number of a graph

The distance d ( x,y ) is the length of the shortest path between a pair of vertices x, y ∊ V ( G ) of a connected graph G = (V, E) . Whereas the detour distance D ( x, y ) is the length of the longest path between x and y in G . Given a subset S ⊆ V ( G ) the set d ( S ) is the multiset of pairwise distances between the vertices of S . Furthermore, D ( S ) is the multiset of pairwise detour distances between the vertices of S . Two subsets of the vertex set of a graph G are said to be homometric if their distance multisets are same. The largest integer h such that there are two disjoint homometric sets of order h in G is the homometric number of G denoted by h ( G ). In this article, we extend the notion of the homometric number of a graph to detour the homometric number of a graph as follows: two subsets of the vertex set of a graph G is detour homometric if the multisets D ( S ) of detour distances determined by them are same. Detour homometric number h D (G) of a graph G is the largest integer h D such that there are two disjoint detour homometric sets of order h D in G. In this article, we explore the notion of detour homometric sets for graphs containing at least on cycle and also propose a potential application of detour homometric sets for automated vehicles.