Untangling a Planar Graph

Goaoc, Xavier; Kratochvı́l, Jan; Okamoto, Yoshio; Shin, Chan-Su; Spillner, Andreas; Wolff, Alexander

doi:10.1007/s00454-008-9130-6

Cited by 25 publications

(39 citation statements)

References 28 publications

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“…Figure 2 PSGE is related to the notion of planar untangling: Given a straight-line drawing of a planar graph, change the embedding of as few vertices as possible in order to obtain a plane drawing. Goaoc et al [20] described an improvement of a result by Bose et al [4] to show that 4 (n + 1)/2 vertices can always be kept in their original positions. 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.…”

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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Section: Introductionmentioning

confidence: 99%

Column planarity and partially-simultaneous geometric embedding

Barba¹,

Evans²,

Hoffmann³

et al. 2017

JGAA

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“…It is NP-hard to compute fix(G, δ) [5,10] and even to approximate it [5]. At the 5th Czech-Slovak Symposium on Combinatorics in Prague in 1998, Mamoru Watanabe asked whether every polygon on n vertices can be turned into a noncrossing polygon by moving at most εn vertices for some constant ε > 0.…”

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confidence: 99%

“…Trees √ n/2 [ 2] 3 √ n − 3 [ 2] Outerplanar graphs √ n/2 [ 5,8] 2 √ n − 1 + 1 [ 5] Planar graphs 4 √ (n + 1)/2 [ 2] √ n − 2 + 1 [ 5] In the following, a drawing of a graph G is an injective mapping from V (G) to a set of points in R 2 such that each edge vw ∈ E(G) is represented by the straight-line segment between the images of v and w. A drawing is plane if no two edges cross.…”

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confidence: 99%

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Untangling Polygons and Graphs

Cibulka

2009

Discrete Comput Geom

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Untangling is a process in which some vertices in a drawing of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C n while keeping (n 2/3 ) vertices fixed.For any connected graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree, and diameter of G. One consequence is that every 3-connected planar graph has a drawing δ such that at most O((n log n) 2/3 ) vertices are fixed in every untangling of δ.

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“…Goaoc et al [4] showed that it is N P-hard to decide whether a given graph can be 1-bend embedded on a given set of points with given vertex-point correspondence. -We show that Geodesic Embeddability on the grid is equivalent to deciding whether the given graph has a rectilinear one-bend drawing on the grid, see Section 2.…”

Section: Introductionmentioning

confidence: 99%

Manhattan-Geodesic Embedding of Planar Graphs

et al. 2010

Self Cite

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Abstract. In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is N P-hard. In contrast, we efficiently solve geodesic polygonization-the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex-point correspondence is given. We show that on the grid, this problem is N P-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.

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Untangling a Planar Graph

Cited by 25 publications

References 28 publications

Column planarity and partially-simultaneous geometric embedding

Column planarity and partially-simultaneous geometric embedding

Untangling Polygons and Graphs

Manhattan-Geodesic Embedding of Planar Graphs

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