On Turn-Regular Orthogonal Representations

TL;DR

This paper studies turn-regular orthogonal graph representations, showing they allow linear-time minimum-area drawings and identifying classes of graphs that always admit such representations, with efficient algorithms for testing their existence.

Contribution

It characterizes graph classes that admit turn-regular orthogonal representations and provides linear or polynomial-time algorithms for constructing or testing these representations.

Findings

Turn-regular representations enable linear-time minimum-area drawings.

Certain biconnected planar graphs always admit turn-regular representations.

Efficient algorithms exist for testing turn-regular representations in specific graph classes.

Abstract

An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignments of vertex and bend coordinates of H. In contrast, finding a minimum-area drawing of H is NP-hard if H is non-turn-regular. This scenario naturally motivates the study of which graphs admit turn-regular orthogonal representations. In this paper we identify notable classes of biconnected planar graphs that always admit such representations, which can be computed in linear time. We also describe a linear-time testing algorithm for trees and provide a polynomial-time algorithm that tests whether a biconnected plane graph with "small" faces has a turn-regular orthogonal representation without bends.