Variants of the Segment Number of a Graph

Okamoto, Yoshio; Ravsky, Alexander; Wolff, Alexander

doi:10.1007/978-3-030-35802-0_33

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…Other related work. Okamoto et al [12] investigated variants of the segment number. For planar graphs in 2D, they allowed bends.…”

Section: Introductionmentioning

confidence: 99%

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

Ina¹,

Klawitter²,

Klemz³

et al. 2022

Preprint

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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, 2trees, and planar 3-trees. It is known that every cubic triconnected planar n-vertex graph (except K4) 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, 2trees, 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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“…Other related work. Okamoto et al [12] investigated variants of the segment number. For planar graphs in 2D, they allowed bends.…”

Section: Introductionmentioning

confidence: 99%

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

Ina¹,

Klawitter²,

Klemz³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

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“…These graphs form a well-structured family of planar graphs having many interesting properties: starting from recognition [7,16] to other graph characteristics [7,24]. They are also used to find the computational complexities of various geometric graph parameters [8,14,45]. However, these graphs are not that well studied.…”

Section: Introductionmentioning

confidence: 99%

Pseudoline arrangement graphs: degree sequences and eccentricities

Das,

Rao,

Sahoo

2021

Preprint

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A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give a simple criterion based on the degree sequence that says whether a degree sequence will have a pseudoline arrangement graph as one of its realizations. In the second part, we study the eccentricities of vertices in such graphs. We observe that the diameter (maximum eccentricity of a vertex in the graph) of any pseudoline arrangement graph on n pseudolines is n − 2. Then we characterize the diametrical vertices (whose eccentricity is equal to the graph diameter) of pseudoline arrangement graphs. These results hold for line arrangement graphs as well.

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