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

Abstract

AbstractThe of a planar graph G is the smallest number of line segments needed for a planar straight-line drawing of G. Dujmović, 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 Dujmović 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.KeywordsVisual complexitySegment numberLower/upper bounds