Simultaneous Embeddability of Two Partitions

TL;DR

This paper explores the conditions under which two partitions of a set can be simultaneously embedded in the plane with regions representing blocks, establishing classifications, characterizations, and computational complexities for each class.

Contribution

It introduces three classes of simultaneous embeddability for partitions, characterizes each class, and analyzes the computational complexity of deciding embeddability.

Findings

Every pair of partitions has a weak simultaneous embedding.

Deciding strong simultaneous embeddability is NP-complete.

Full simultaneous embeddability can be tested in linear time.

Abstract

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.