# Planar Graphs: Random Walks and Bipartiteness Testing

Conference: The 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011).
Journal: Random Structures & Algorithms 55(1), 2019.

Abstract: We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time: for every planar graph $G$ and $\varepsilon>0$, we can distinguish in constant time between the case that $G$ is bipartite and the case that $G$ is $\varepsilon$-far from bipartite. The previous bound for this class of graphs was $\tilde{O}(\sqrt{n})$, where $n$ is the number of vertices, and the constant-time testability was only known for planar graphs with bounded degree. Our approach extends to arbitrary minor-free graphs.

Our algorithm is based on random walks. The challenge here is to analyze random walks for graphs that have good separators, i.e., bad expansion. Standard techniques that use a fast convergence of random walks to a uniform distribution do not work in this case. Informally, our approach is to self-reduce the problem of finding an odd-length cycle in a multigraph $G$ induced by a collection of cycles to the same problem on another multigraph $G'$ induced by a set of shorter odd-length cycles, in such a way that when a random walk finds a cycle in $G'$ with probability $p > 0$, then it does so in $G$ with probability $\lambda(p)>0$. This reduction is applied until the cycles collapse to self-loops, in which case they can be easily detected.

While the analysis presented in this paper applies only to testing bipartiteness, we believe that the techniques developed will find applications to testing other properties in arbitrary planar (or minor-free) graphs, in a similar way as in the past the advances in testing bipartiteness led to the development of testing algorithms for more complex graph properties.

Full version: [arXiv]
BibTeX: [DBLP]