## Testing properties in arbitrary planar graphs via random walks

by Artur Czumaj, Morteza Monemizadeh, Krzysztof Onak, and Christian Sohler.

We study the testability of hereditary properties in arbitrary planar graphs. Our main result is a constant time testing of bipartiteness in arbitrary planar graphs. The previous bound for this class of graphs was $O^{\ast}(\sqrt{n})$, and the constant-time testability was only known for planar graphs with {\bf bounded degree}. Previously used transformations of unbounded-degree sparse graphs into bounded-degree sparse graphs cannot be used to reduce the problem to the testability of bounded-degree planar graphs. Our approach extends to arbitrary minor-free graphs. Our algorithm is based on random walks. The challenge here is to analyze random walks for a class of graphs that has good separators, i.e., bad expansion. Standard techniques that use a fast convergence to a uniform distribution do not work in this case. Informally, our analysis technique self-reduces the problem of finding an odd length cycle in a multigraph $G$ induced by a collection of cycles to another multigraph $G'$ induced by a set of shorter odd-length cycles, in such a way that when a random walks finds a cycle in $G'$ with probability p, then it does so with probability $f(p)$ in $G$. This reduction is applied until the cycles collapse to selfloops that can be easily detected.