Planar Graphs: Random Walks and Bipartiteness Testing

TL;DR

This paper presents a constant-time property testing algorithm for bipartiteness in planar graphs using random walks, significantly improving previous bounds and extending to minor-free graphs.

Contribution

It introduces a novel constant-time bipartiteness testing algorithm for planar graphs based on random walks, surpassing prior bounds and applicable to minor-free graphs.

Findings

Bipartiteness can be tested in constant time in planar graphs.

The algorithm extends to arbitrary minor-free graphs.

It leverages properties of planar graphs and random walks for efficient testing.

Abstract

We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time testability was only known for planar graphs with bounded degree. Our algorithm is based on random walks. Since planar graphs have good separators, i.e., bad expansion, our analysis diverges from standard techniques that involve the fast convergence of random walks on expanders. We reduce the problem to the task of detecting an odd-parity cycle in a multigraph induced by constant-length cycles. We iteratively reduce the length of cycles while preserving the detection probability, until the multigraph collapses to a collection of easily discoverable self-loops. Our approach extends to arbitrary minor-free graphs. We also believe that our techniques will find applications to testing other properties in arbitrary minor-free graphs.