平面图的随机游动和二分性检验
Planar Graphs: Random Walks and Bipartiteness Testing
摘要：
本文首先研究了任意平面图的性质测试问题。我们证明了可以在常数时间内测试二分性，从而改进了之前针对$n$顶点上的图的$\tilde{O}(\sqrt{n})$的界。常数时间可测性只存在于度有界的平面图中。我们的算法是基于随机游动。由于平面图具有良好的分隔符，即，坏的扩展，我们的分析偏离标准技术，涉及快速收敛的随机游动的扩展。我们减少了这个问题的任务，检测奇奇偶校验周期在一个多重图引起的恒定长度的周期。我们迭代地减少循环的长度，同时保持检测概率，直到多重图崩溃到一个容易折叠的自循环的集合。我们的方法扩展到任意小自由图。我们也相信，我们的技术将找到应用程序，以测试其他属性在任意小的自由图。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
