Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

Steven Chaplick; Giordano Da Lozzo; Emilio Di Giacomo; Giuseppe Liotta; Fabrizio Montecchiani

doi:10.48550/arXiv.2105.08124

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

Steven Chaplick
, Giordano Da Lozzo
, Emilio Di Giacomo
, Giuseppe Liotta
, Fabrizio Montecchiani

Research output: Working paper / Preprint › Preprint

Abstract

The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of degree $\Delta$. This upper bound has been improved to $O(\Delta^5)$ if $G$ has treewidth three, and to $O(\Delta)$ if $G$ has treewidth two. In this paper we prove $psn(G) \in \Theta(\Delta)$ when $G$ is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $O(\Delta^2)$ slopes suffice for nested pseudotrees.

Original language	English
DOIs	https://doi.org/10.48550/arXiv.2105.08124
Publication status	Published - 17 May 2021

Keywords

cs.CG

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10.48550/arXiv.2105.08124Licence: CC BY

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees
Chaplick, S., Da lozzo, G., Di giacomo, E., Liotta, G. & Montecchiani, F., Aug 2024, In: Algorithmica. 86, 8, p. 2413-2447 35 p.
Research output: Contribution to journal › Article › Academic › peer-review

Open Access
Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees
Chaplick, S., Da Lozzo, G., Di Giacomo, E., Liotta, G. & Montecchiani, F., 2021, Algorithms and Data Structures - 17th International Symposium, WADS 2021, Proceedings. Lubiw, A. & Salavatipour, M. (eds.). Springer Nature, Vol. 12808. p. 271-285 15 p. (Lecture Notes in Computer Science, Vol. 12808).
Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

Open Access

Cite this

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title = "Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees",

abstract = "The \$\textbackslash{}textit\{planar slope number\}\$ \$psn(G)\$ of a planar graph \$G\$ is the minimum number of edge slopes in a planar straight-line drawing of \$G\$. It is known that \$psn(G) \textbackslash{}in O(c\textasciicircum{}\textbackslash{}Delta)\$ for every planar graph \$G\$ of degree \$\textbackslash{}Delta\$. This upper bound has been improved to \$O(\textbackslash{}Delta\textasciicircum{}5)\$ if \$G\$ has treewidth three, and to \$O(\textbackslash{}Delta)\$ if \$G\$ has treewidth two. In this paper we prove \$psn(G) \textbackslash{}in \textbackslash{}Theta(\textbackslash{}Delta)\$ when \$G\$ is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that \$O(\textbackslash{}Delta\textasciicircum{}2)\$ slopes suffice for nested pseudotrees.",

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AU - Montecchiani, Fabrizio

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N2 - The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of degree $\Delta$. This upper bound has been improved to $O(\Delta^5)$ if $G$ has treewidth three, and to $O(\Delta)$ if $G$ has treewidth two. In this paper we prove $psn(G) \in \Theta(\Delta)$ when $G$ is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $O(\Delta^2)$ slopes suffice for nested pseudotrees.

AB - The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of degree $\Delta$. This upper bound has been improved to $O(\Delta^5)$ if $G$ has treewidth three, and to $O(\Delta)$ if $G$ has treewidth two. In this paper we prove $psn(G) \in \Theta(\Delta)$ when $G$ is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $O(\Delta^2)$ slopes suffice for nested pseudotrees.

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BT - Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

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Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

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