The Utility of Untangling

Dujmović, Vida

doi:10.1007/978-3-319-27261-0_27

Cited by 6 publications

(13 citation statements)

References 39 publications

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“…Theorem 1 then implies Corollary 1 since every free collinear set is also a free set. This fact, which implies that v(G) = v(G), has been observed by several authors [5,9,13,15]. To see it, let X = {(x 1 , y 1 ), .…”

Section: Introductionsupporting

confidence: 55%

“…Corollary 3 is almost tight due to the O( n log 3 n) upper bound for triconnected cubic planar graphs of diameter O(log n) [8]. Corollary 3 cannot be extended to all boundeddegree planar graphs, see [13,20] for reasons why. Da Lozzo et al [9] also proved that planar graphs of treewidth at least k have Ω(k 2 )-size collinear sets.…”

Section: Applications and Related Workmentioning

confidence: 99%

“…A universal point subset for a set G of n-vertex planar graphs is a set P of k ≤ n points in the plane such that, for every G ∈ G, there is a plane straight-line drawing of G in which k vertices of G are mapped to the k points in P . Every set of n points in general position is a universal point subset for n-vertex outerplanar graphs [4,7,16]; every set of n−3 8 points in the plane is a universal point subset for the n-vertex planar graphs of treewidth at most three [9]; and, every set of n 2 points in the plane is a universal point subset for the n-vertex planar graphs [13].…”

Section: Applications and Related Workmentioning

confidence: 99%

“…A set of vertices S ⊆ V (G) in a planar graph G is a free set if for any set of points X in the plane with |X| = |S|, G has a plane straight-line drawing in which the vertices of S are mapped to the points in X. Free sets are useful tools in graph drawing and related areas and have been used to settle problems in untangling [5,9,13,22], column planarity [9,13], universal point subsets [9,13], and partial simultaneous geometric drawings [13]. A set of vertices S ⊆ V (G) in a planar graph G is a collinear set if G has a plane straightline drawing in which all vertices in S are mapped to a single line, see Figure 1.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionsupporting

confidence: 55%

Section: Applications and Related Workmentioning

confidence: 99%

Section: Applications and Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Every Collinear Set in a Planar Graph is Free

Dujmović

Frati

Morin

et al. 2020

Discrete Comput Geom

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“…It would be interesting to see whether polynomial area is sufficient. Recently, Dujmović [12] showed that every planar graph has a column planar subset of size at least n/2. No upper bounds other than the one from Theorem 2 are known.…”

Section: Discussionmentioning

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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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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The Utility of Untangling

Cited by 6 publications

References 39 publications

Every Collinear Set in a Planar Graph is Free

Every Collinear Set in a Planar Graph is Free

Column planarity and partially-simultaneous geometric embedding

Drawing Planar Graphs with Many Collinear Vertices

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