@article{Hayat2025, 
author = {Sakander Hayat and Bagus Imanda and Asad Khan and Mohammed J. F. Alenazi},
title = {Three infinite families of Hamilton-connected convex polytopes and their detour index},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {5},
pages = {12343-12387},
keywords = {graph, Hamiltonian path, Hamiltonian cycle, Hamilton-connected graph, detour index, NP-complete problems, convex polytopes},
url = {https://www.sciopen.com/article/10.3934/math.2025559},
doi = {10.3934/math.2025559},
abstract = {A path in a graph encompassing its whole vertex set is called Hamiltonian. Such a path with sharing the same initial and terminal vertices is called a Hamiltonian cycle. A graph comprising a Hamiltonian path (resp. cycle) is said to be traceable (resp. Hamiltonian). Graphs possessing Hamiltonian paths between every pair of their vertices are said to be Hamilton-connected. The computational complexity of evaluating a graph to be Hamilton-connected is NP-complete. A detour is the longest path in a graph. The detour index is the sum of the length of detours between every unordered pair of vertices. Computing the detour index of a graph is an NP-complete problem as well. A finite subset    P  ⊂            R        ε   is called a convex polytope if    P is a convex hull. In this paper, we devised two distinct methods to prove a graph to be Hamilton-connected and employed these methods to construct some infinite families of Hamilton-connected convex polytopes. The convex polytope        B    ε   has been shown to be non-Hamilton-connected in the literature. We showed that the existing proof for        B    ε   is false and showed that this family is, in fact, Hamilton-connected. The paper is concluded with study implications followed by some future directions.}
}