Unit-length embedding of cycles and paths on grid graphs

TL;DR

This paper characterizes when cycles and paths of specific lengths can be embedded in grid graphs, providing algorithms with linear or quadratic time complexity, and extends results to 3D grids and hamiltonian solid grids.

Contribution

It offers necessary and sufficient conditions for embedding paths and cycles of given lengths in grid graphs, along with efficient algorithms and extensions to 3D and solid grid graphs.

Findings

Conditions for embedding cycles and paths of given length

Algorithms with O(k) and O(k^2) running times for finding such paths and cycles

Extension of methods to 3D grids and hamiltonian solid grid graphs

Abstract

Although there are very algorithms for embedding graphs on unbounded grids, only few results on embedding or drawing graphs on restricted grids has been published. In this work, we consider the problem of embedding paths and cycles on grid graphs. We give the necessary and sufficient conditions for the existence of cycles of given length $k$ and paths of given length $k$ between two given vertices in $n$-vertex rectangular grid graphs and introduce two algorithms with running times O$(k)$ and O$(k^2)$ for finding respectively such cycles and paths. Also, we extend our results to $m\times n\times o$ 3D grids. Our method for finding cycle of length $k$ in rectangular grid graphs also introduces a linear-time algorithm for finding cycles of a given length $k$ in hamiltonian solid grid graphs.