Publication

Abstract : 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.