The Segment Number: Algorithms and Universal Lower Bounds for Some Classes of Planar Graphs

TL;DR

This paper investigates the segment number of various classes of planar graphs, providing new upper and lower bounds, and introduces algorithms that optimize the visual complexity of graph drawings.

Contribution

It presents tight bounds for the segment number of triconnected planar 4-regular graphs and establishes the first linear universal lower bounds for several graph classes.

Findings

Triconnected planar 4-regular graphs can be drawn with at most n+3 segments.

Linear universal lower bounds are proven for outerpaths, outerplanar graphs, 2-trees, and planar 3-trees.

Existing algorithms are shown to be constant-factor approximations for these classes.

Abstract

The segment number of a planar graph $G$ is the smallest number of line segments needed for a planar straight-line drawing of $G$. Dujmovi\'c, Eppstein, Suderman, and Wood [CGTA'07] introduced this measure for the visual complexity of graphs. There are optimal algorithms for trees and worst-case optimal algorithms for outerplanar graphs, 2-trees, and planar 3-trees. It is known that every cubic triconnected planar $n$-vertex graph (except $K_4$) has segment number $n/2+3$, which is the only known universal lower bound for a meaningful class of planar graphs. We show that every triconnected planar 4-regular graph can be drawn using at most $n+3$ segments. This bound is tight up to an additive constant, improves a previous upper bound of $7n/4+2$ implied by a more general result of Dujmovi\'c et al., and supplements the result for cubic graphs. We also give a simple optimal algorithm for cactus graphs, generalizing the above-mentioned result for trees. We prove the first linear universal lower bounds for outerpaths, maximal outerplanar graphs, 2-trees, and planar 3-trees. This shows that the existing algorithms for these graph classes are constant-factor approximations. For maximal outerpaths, our bound is best possible and can be generalized to circular arcs.