NDLI: Plane 3-Trees: Embeddability and Approximation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Durocher, Stephane Mondal, Debajyoti
Copyright Year	2015
Abstract	We give an $O(n\log ^3 n)$-time linear-space algorithm that, given a plane 3-tree $G$ with $n$ vertices and a set $S$ of $n$ points in the plane, determines whether $G$ has a point-set embedding on $S$ (i.e., a planar straight-line drawing of $G$ where each vertex is mapped to a distinct point of $S$), improving the $O(n^{4/3+\varepsilon})$-time $O(n^{4/3})$-space algorithm of Moosa and Rahman [Lecture Notes in Comput. Sci. 6842, Springer, New York, 2011, pp. 204--212]. Given an arbitrary plane graph $G$ and a point set $S$, Kaufmann and Wiese [J. Graph Algorithms Appl., 6 (2002), pp. 115--129] gave an algorithm to compute 2-bend point-set embeddings of $G$ on $S$. Later, Di Giacomo and Liotta [Lecture Notes in Comput. Sci. 5942, Springer, New York, 2010, pp. 35--46] showed how such a drawing can be computed using $O(W^3)$ area, where $W$ is the length of the longest edge of the bounding box of $S$. Their algorithm uses $O(W^3)$ area even when the input graphs are restricted to plane 3-trees. We introduce new techniques for computing $2$-bend point-set embeddings of plane 3-trees that take only $O(W^2)$ area. We also give approximation algorithms for point-set embeddings of plane $3$-trees. Our results on 2-bend point-set embeddings and approximate point-set embeddings hold for partial plane $3$-trees (e.g., series-parallel graphs and Halin graphs).
Starting Page	405
Ending Page	420
Page Count	16
File Format	PDF
ISSN	08954801
DOI	10.1137/140964710
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	29
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-02-24
Access Restriction	Subscribed
Subject Keyword	Planar graphs; geometric and topological aspects of graph theory approximation point-set embedding graph drawing plane 3-tree Graph theory
Content Type	Text
Resource Type	Article
Subject	Mathematics

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