The Hamiltonicity, Hamiltonian Connectivity, and Longest (s, t)-path of L-shaped Supergrid Graphs

TL;DR

This paper investigates Hamiltonian properties of L-shaped supergrid graphs, proving conditions for Hamiltonicity and connectivity, and provides a linear-time algorithm for the longest (s, t)-path, with applications in manufacturing.

Contribution

It establishes new conditions for Hamiltonian cycle and connectivity in L-shaped supergrid graphs and introduces an efficient algorithm for longest path computation.

Findings

Almost all L-shaped supergrid graphs contain a Hamiltonian cycle.

Most L-shaped supergrid graphs are Hamiltonian connected.

A linear-time algorithm for longest (s, t)-path is developed.

Abstract

Supergrid graphs contain grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for general supergrid graphs were known to be NP-complete. A graph is called Hamiltonian if it contains a Hamiltonian cycle, and is said to be Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it. In this paper, we first prove that every L-shaped supergrid graph always contains a Hamiltonian cycle except one trivial condition. We then verify the Hamiltonian connectivity of L-shaped supergrid graphs except few conditions. The Hamiltonicity and Hamiltonian connectivity of L-shaped supergrid graphs can be applied to compute the minimum trace of computerized embroidery machine and 3D printer when a L-like object is printed. Finally, we present a linear-time algorithm to compute the longest (s, t)-path of L-shaped supergrid graph given two distinct vertices s and t.