Abstract
A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.
Highlights
Introduction and PreliminariesAll graphs in this paper are simple, loopless, finite, and connected
A Hamiltonian path PH(x, y) between vertices x and y is the one covering the entire graph without missing any vertex
Kratica et al [18] studied the strong metric dimension of certain infinite families of convex polytopes by constructing their doubly resolving sets. e fault-tolerant metric dimension of convex polytopes was studied by Raza et al [19]
Summary
All graphs in this paper are simple, loopless, finite, and connected. A graph G is an ordered pair G (V(G), E(G)) with. Graphs comprising Hamiltonian paths between every pair of its vertices are called Hamilton-connected. Imran et al [12,13,14,15] computed the minimum metric dimension of various infinite families of convex polytopes. Kratica et al [18] studied the strong metric dimension of certain infinite families of convex polytopes by constructing their doubly resolving sets. Mixed metric dimension) of convex polytopes was studied by Raza et al [19] Hayat et al [24] studied Hamilton-connectivity and detour index in convex polytopes. Karbasioun et al [35] studied the applications of the detour index in infinite families of nanostar dendrimers. Abdullah and Omar [38] introduced the restricted edge version of the detour index and studied it for some families of graphs. With left equality if and only if G S], and right inequality holds if and only if G is Hamilton-connected
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