Analogies between the Crossing Number and the Tangle Crossing Number

Anderson, Roger C.; Bai, Shuliang; Barrera-Cruz, Fidel; Czabarka, Éva; Lozzo, Giordano Da; Hobson, Natalie; Lin, Jephian C.-H.; Mohr, Austin; Smith, Heather; Székely, László Á.; Whitlatch, Hays

doi:10.37236/7581

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…Before we answer (1), we start with a definition. While a special case of this definition is sufficient for answering (1), the extra generality will be useful later in Section 5.…”

Section: Preserving Subtanglegram Planarity and Reducing Crossingsmentioning

confidence: 99%

“…Prior to this, Lozano et al constructed their Untangle Algorithm for drawing a planar layout of a planar tanglegram [15]. Anderson et al recently proved that removing a between-tree edge (t i , s φ(i) ) from a tanglegram reduces the crossing number by at most n − 3, and they produced a family of tanglegrams to show that this bound is sharp [1]. They also found that the maximum crossing number over all tanglegrams asymptotically approaches 1 2 n 2 , where n is the number of leaves in each tree.…”

Section: Introductionmentioning

confidence: 99%

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Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Liu

2023

Electron. J. Combin.

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Tanglegrams are formed by taking two rooted binary trees $T$ and $S$ with the same number of leaves and uniquely matching each leaf in $T$ with a leaf in $S$. They are usually represented using layouts that embed the trees and matching in the plane. Given the numerous ways to construct a layout, one problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels a similar problem involving drawings of graphs, where a common approach is to insert edges into a planar subgraph. In this paper, we explore inserting edges into a planar tanglegram. Previous results on planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm for finding a planar layout. We build on these results and characterize all planar layouts of a planar tanglegram. We then apply this characterization to construct a quadratic-time algorithm that inserts a single edge optimally. Finally, we generalize some results to multiple edge insertion.

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“…Before we answer (1), we start with a definition. While a special case of this definition is sufficient for answering (1), the extra generality will be useful later in Section 5.…”

Section: Preserving Subtanglegram Planarity and Reducing Crossingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Liu

2023

Electron. J. Combin.

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show abstract

“…constructed their Untangle Algorithm for drawing a planar layout of a planar tanglegram [15]. Anderson et al recently proved that removing a between-tree edge (t i , s φ(i) ) from a tanglegram reduces crossing number by at most n − 3, and they produced a family of tanglegrams to show that this bound is sharp [1]. They also found that the maximum crossing number over all tanglegrams asymptotically approaches 1 2 n 2 , where n is the number of leaves in each tree.…”

Section: Introductionmentioning

confidence: 99%

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Liu¹

2022

Preprint

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Tanglegrams are formed by taking two rooted binary trees T and S with the same number of leaves and uniquely matching each leaf in T with a leaf in S. They are usually represented using layouts, which embed the trees and the matching of the leaves into the plane as in Figure 1. Given the numerous ways to construct a layout, one problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels a similar problem involving drawings of graphs, where a common approach is to insert edges into a planar subgraph. In this paper, we will explore inserting edges into a planar tanglegram. Previous results on planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm for drawing a planar layout [5,16,15]. We start by building on these results and characterizing all planar layouts of a planar tanglegram. We then apply this characterization to construct a quadratic-time algorithm that inserts a single edge optimally. Finally, we generalize some results to multiple edge insertion.

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