Hamiltonian Cycles in Linear-Convex Supergrid Graphs

TL;DR

This paper proves that all 2-connected, linear-convex supergrid graphs have Hamiltonian cycles, expanding understanding of Hamiltonian properties in specific subclasses of supergrid graphs.

Contribution

It establishes that 2-connected, linear-convex supergrid graphs always contain a Hamiltonian cycle, a new result in the study of supergrid graph properties.

Findings

Any 2-connected, linear-convex supergrid graph is locally connected.

Any 2-connected, linear-convex supergrid graph contains a Hamiltonian cycle.

The class of linear-convex supergrid graphs has Hamiltonian cycles under certain connectivity conditions.

Abstract

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. In the past, we have shown that the Hamiltonian cycle problem for supergrid graphs is also NP-complete. The Hamiltonian cycle problem on supergrid graphs can be applied to control the stitching trace of computerized sewing machines. In this paper, we will study the Hamiltonian cycle property of linear-convex supergrid graphs which form a subclass of supergrid graphs. A connected graph is called $k$-connected if there are $k$ vertex-disjoint paths between every pair of vertices, and is called locally connected if the neighbors of each vertex in it form a connected subgraph. In this paper, we first show that any 2-connected, linear-convex supergrid graph is locally connected. We then prove that any 2-connected, linear-convex supergrid graph contains a Hamiltonian cycle.