嵌入循环c-图的c-平面性
C-planarity of Embedded Cyclic c-Graphs
摘要：
我们表明，c-平面性是可解的平方时间的平坦的聚类图与三个集群，如果底层图的组合嵌入是固定的。用简单的图论术语来说，我们的结果可以看作如下。给定一个图$G$，其顶点集被划分为嵌入在2-球面上的三个部分，我们的算法决定我们是否可以通过添加边而不创建边交叉来扩充$G$，使得在所得到的球面图中，每个部分的顶点诱导一个连通子图。我们着手减少的问题，测试存在的一个完美的匹配在平面二分图。我们制定我们的结果在一个稍微更一般的设置循环聚类图，即，通过收缩每个簇而获得的简单图是一个循环，其中我们忽略了环和多边。
英文摘要：
We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph $G$ with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment $G$ by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.
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。
