Upper Bound Constructions for Untangling Planar Geometric Graphs

Cano, Javier; Tóth, Csaba D.; Urrutia, Jorge

doi:10.1137/130924172

Cited by 6 publications

(9 citation statements)

References 13 publications

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“…The goal is to do so while keeping fixed (i.e., not changing the location of) as many vertices as possible. Several papers have studied the untangling problem [18,4,7,3,11,15,20]. It is known that general n-vertex planar graphs can be untangled while keeping Ω(n 0.25 ) vertices fixed [3] and that there are n-vertex planar graphs that cannot be untangled while keeping Ω(n 0.4948 ) vertices fixed [4].…”

Section: Introductionmentioning

confidence: 99%

“…Several papers have studied the untangling problem [18,4,7,3,11,15,20]. It is known that general n-vertex planar graphs can be untangled while keeping Ω(n 0.25 ) vertices fixed [3] and that there are n-vertex planar graphs that cannot be untangled while keeping Ω(n 0.4948 ) vertices fixed [4]. Asymptotically tight bounds are known for paths [7], trees [11], outerplanar graphs [11], and planar graphs of treewidth two [20].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Drawing Planar Graphs with Many Collinear Vertices

Lozzo

Dujmović

Frati

et al. 2016

Lecture Notes in Computer Science

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Consider the following problem: Given a planar graph G, what is the maximum number p such that G has a planar straight-line drawing with p collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every n-vertex planar graph has a planar straight-line drawing with Ω( √ n) collinear vertices; for every n, there is an n-vertex planar graph whose every planar straight-line drawing has O(n σ ) collinear vertices, where σ < 0.986; every n-vertex planar graph of treewidth at most two has a planar straight-line drawing with Θ(n) collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295-306]. Similar results are not possible for all bounded treewidth planar graphs or for all bounded degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Drawing Planar Graphs with Many Collinear Vertices

Lozzo

Dujmović

Frati

et al. 2016

Lecture Notes in Computer Science

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“…This has almost been matched by an Ω(n 2/3 ) lower bound by Cibulka [8]. Several papers have studied the untangling problem [5,6,8,15,17,21,22]. Asymptotically tight bounds are known for paths [8], trees [15], outerplanar graphs [15], and planar graphs of treewidth two and three [9,22].…”

Section: Applications and Related Workmentioning

confidence: 96%

“…For general planar graphs there is still a large gap. Namely, it is known that every planar graph can be untangled while keeping Ω(n 0.25 ) vertices fixed [5] (this answered a question by Pach and Tardos [21]) and that there are planar graphs that cannot be untangled while keeping Ω(n 0.4948 ) vertices fixed [6]. Theorem 1 can help close this gap, whenever a good bound on collinear sets is known.…”

Section: Applications and Related Workmentioning

confidence: 99%

Every Collinear Set in a Planar Graph is Free

Dujmović

Frati

Morin

et al. 2020

Discrete Comput Geom

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We show that if a planar graph G has a plane straight-line drawing in which a subset S of its vertices are collinear, then for any set of points, X, in the plane with |X| = |S|, there is a plane straight-line drawing of G in which the vertices in S are mapped to the points in X. This solves an open problem posed by Ravsky and Verbitsky in 2008. In their terminology, we show that every collinear set is free.This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings.

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“…Since we can simply take any plane embedding of G 1 , use the same embedding for G 2 and then untangle G 2 , it immediately follows that every two planar graphs on n vertices admit a 4 (n + 1)/2-PSGE. In the other direction, Cano et al [7] showed the existence of a graph and a straight-line drawing of this graph such that in any planar untangling, the embedding of all but O(n .4965 ) vertices has to change. However, an upper bound for untangling does not have direct implications for PSGE because untangling is more restrictive compared to PSGE: In the untangling problem the positions of all vertices, in particular, of those that remain fixed, are given, whereas in PSGE one may select suitable positions for those vertices.…”

Section: Introductionmentioning

confidence: 99%

Column planarity and partially-simultaneous geometric embedding

Barba¹,

Evans²,

Hoffmann³

et al. 2017

JGAA

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We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straightline drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for the maximum size of column planar subsets of trees: every tree on n vertices contains a column planar set of size at least 14n/17 and for any > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + )n. In addition, we show that every outerplanar graph has a column planar set of size at least n/2.We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (PSGE). A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs, which are PSGEs in which at least k vertices are mapped to the same point for both graphs. In particular, we show that every two trees on n vertices admit an 11n/17-PSGE and every two outerplanar graphs admit an n/4-PSGE.

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Upper Bound Constructions for Untangling Planar Geometric Graphs

Cited by 6 publications

References 13 publications

Drawing Planar Graphs with Many Collinear Vertices

Drawing Planar Graphs with Many Collinear Vertices

Every Collinear Set in a Planar Graph is Free

Column planarity and partially-simultaneous geometric embedding

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