平面性测试再探

摘要：

平面性测试是确定给定图是否是平面图的问题，而平面嵌入是相应的构造问题。Allender和Mahajan借助于Reingold的结果，确定了这类问题的有界空间复杂度恰好为Logspace。不幸的是，该算法是相当令人生畏的，概括起来说，有界亏格的情况似乎是一个很高的要求。在这项工作中，我们提出了一个简单的平面嵌入算法运行在对数空间。我们希望这个算法将更易于推广。该算法是基于3-连通平面图具有唯一嵌入、Tutte关于圈冲突图的判别准则的一个变体以及圈基的显式变化这一事实。对于平面图。我们还提出了一个对数空间算法来找到平面性的障碍，即一个Kuratowski未成年人，如果图是非平面的。据我们所知，这是这个问题的第一个对数空间算法。

英文摘要：

Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem. The bounded space complexity of these problems has been determined to be exactly Logspace by Allender and Mahajan with the aid of Reingold's result. Unfortunately, the algorithm is quite daunting and generalizing it to say, the bounded genus case seems a tall order. In this work, we present a simple planar embedding algorithm running in logspace. We hope this algorithm will be more amenable to generalization. The algorithm is based on the fact that 3-connected planar graphs have a unique embedding, a variant of Tutte's criterion on conflict graphs of cycles and an explicit change of cycle basis.% for planar graphs. We also present a logspace algorithm to find obstacles to planarity, viz. a Kuratowski minor, if the graph is non-planar. To the best of our knowledge this is the first logspace algorithm for this problem.

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。