Embedding the dual complex of hyper-rectangular partitions

TL;DR

This paper investigates the complexity of embedding the dual graph of hyper-rectangular partitions, proving NP-completeness for the general case and exploring conditions for nice embeddings in both 2D and higher dimensions.

Contribution

It establishes the NP-completeness of deciding the existence of a straight-line dual embedding for rectangular partitions and analyzes conditions for centered embeddings in multiple dimensions.

Findings

Deciding the existence of a nice dual drawing is NP-complete.

Centered embeddings can be characterized under certain conditions.

The problem extends to higher-dimensional rectangular partitions.

Abstract

A rectangular partition is the partition of an (axis-aligned) rectangle into interior-disjoint rectangles. We ask whether a rectangular partition permits a "nice" drawing of its dual, that is, a straight-line embedding of it such that each dual vertex is placed into the rectangle that it represents. We show that deciding whether such a drawing exists is NP-complete. Moreover, we consider the drawing where a vertex is placed in the center of the represented rectangle and consider sufficient conditions for this drawing to be nice. This question is studied both in the plane and for the higher-dimensional generalization of rectangular partitions.