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