两个划分的同时可嵌入性
Simultaneous Embeddability of Two Partitions
摘要：
我们研究了同一底层集合的一对划分到不相交块中的同时嵌入性。集合中的每个元素都映射到平面中的一个点，两个分区中的任何一个的每个块都映射到一个区域，该区域恰好包含属于块中元素的那些点，并且由简单的闭合曲线限定。我们建立了三种主要的同时可嵌入性（弱可嵌入性、强可嵌入性和完全可嵌入性），它们的不同之处在于允许不同块区域相交的形式良好性条件越来越严格。我们发现，这些同时嵌入类是密切相关的超图的不同平面性概念。对于每个可嵌入类，我们给出了一个完整的特征。我们证明了（i）每对划分都有一个弱同时嵌入，（ii）判定强同时嵌入的存在性是NP-完全的，（iii）完全同时嵌入的存在性可以在线性时间内检验.
英文摘要：
We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.
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