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Monotone Drawings of Planar Graphs
[chapter]
2002
Lecture Notes in Computer Science
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Let G be a graph drawn in the plane so that its edges are represented by x-monotone curves, any pair of which cross an even number of times. ...
We show that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments. ...
For any x-monotone even drawing of a connected graph, there is an equivalent x-monotone drawing in which no two edges cross each other and the x-coordinates of the corresponding vertices are the same. ...
doi:10.1007/3-540-36136-7_57
fatcat:5iuh2z7kf5hsjk6hkrmtnlfkli
Monotone drawings of planar graphs
[article]
2011
arXiv
pre-print
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Let G be a graph drawn in the plane so that its edges are represented by x-monotone curves, any pair of which cross an even number of times. ...
We show that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments. ...
For any x-monotone even drawing of a connected graph, there is an equivalent x-monotone drawing in which no two edges cross each other and the x-coordinates of the corresponding vertices are the same. ...
arXiv:1101.0967v1
fatcat:q72qup7mrzdcdoxoegsv6nd3am
Monotone drawings of planar graphs
2004
Journal of Graph Theory
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Let G be a graph drawn in the plane so that its edges are represented by x-monotone curves, any pair of which cross an even number of times. ...
We show that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments. ...
For any x-monotone even drawing of a connected graph, there is an equivalent x-monotone drawing in which no two edges cross each other and the x-coordinates of the corresponding vertices are the same. ...
doi:10.1002/jgt.10168
fatcat:2hzsmn4us5h7hcc6x6dwzy6adu
Monotone Drawings of k-Inner Planar Graphs
[article]
2018
arXiv
pre-print
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An outerplanar graph is a 0-inner planar graph. In this paper, we show how to construct a monotone drawing of a k-inner planar graph on a 2(k+1)n × 2(k+1)n grid. ...
In the special case of an outerplanar graph, we can produce a planar monotone drawing on a n × n grid, improving previously known results. ...
Monotone drawing of k-inner planar graphs 1: procedure k-InnerPlanarMonotoneDrawing(G) 2: Input: An n-vertex planar graph G and an embedding of G with k inner vertices. 3: Output: A monotone planar drawing ...
arXiv:1808.06892v2
fatcat:d2hkcmoaerecraznc77dkmrztq
Monotone Grid Drawings of Planar Graphs
[chapter]
2014
Lecture Notes in Computer Science
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A monotone drawing of a planar graph G is a planar straightline drawing of G where a monotone path exists between every pair of vertices of G in some direction. ...
A monotone drawing of a planar graph is a monotone grid drawing if every vertex in the drawing is drawn on a grid point. ...
We call a monotone drawing of a planar graph a monotone grid drawing if every vertex is drawn on a grid point. ...
doi:10.1007/978-3-319-08016-1_10
fatcat:fmizli4dq5ghjly7a7kcntnb6q
Strongly Monotone Drawings of Planar Graphs
[article]
2016
arXiv
pre-print
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We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. ...
The drawings of 3-connected planar graphs are based on primal-dual circle packings. ...
Every 3-connected planar graph has a strongly monotone drawing. Proof. We show that the straight-line drawing corresponding to a primal-dual circle packing of a graph G is already strongly monotone. ...
arXiv:1601.01598v1
fatcat:3hl7sgjhtbfdxniaf7prt3a4wa
Monotone Grid Drawings of Planar Graphs
[article]
2013
arXiv
pre-print
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A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. ...
A monotone drawing of a planar graph is a monotone grid drawing if every vertex in the drawing is drawn on a grid point. ...
We call a monotone drawing of a planar graph a monotone grid drawing if every vertex is drawn on a grid point. ...
arXiv:1310.6084v1
fatcat:vax3ekavn5dvznmsrxo6a365xy
Strongly Monotone Drawings of Planar Graphs
2016
International Symposium on Computational Geometry
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We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. ...
The drawings of 3-connected planar graphs are based on primal-dual circle packings. ...
1 Every 3-connected planar graph has a strongly monotone drawing. ...
doi:10.4230/lipics.socg.2016.37
dblp:conf/compgeom/FelsnerIKKMS16
fatcat:2fa3uyreofa4lniu5oaie2warm
Strongly Monotone Drawings of Planar Graphs *
unpublished
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We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. ...
The drawings of 3-connected planar graphs are based on primal-dual circle packings. ...
Every 3-connected planar graph has a strongly monotone drawing. Proof. We show that the straight-line drawing corresponding to a primal-dual circle packing of a graph G is already strongly monotone. ...
fatcat:46xipw6jwjabng4dkonx2ws4ei
Height-Preserving Transformations of Planar Graph Drawings
[chapter]
2014
Lecture Notes in Computer Science
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There are numerous styles of planar graph drawings, such as straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. ...
Given a planar drawing in one of these styles, can it be converted it to another style while keeping the height unchanged? ...
The graph in Fig. 3 has a planar poly-line drawing on 6 rows, but no planar y-monotone orthogonal drawing on 6 rows. Proof. ...
doi:10.1007/978-3-662-45803-7_32
fatcat:i35d2e6i4batdkcmgp7aai2bm4
Monotone Drawings of Graphs with Fixed Embedding
[chapter]
2012
Lecture Notes in Computer Science
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In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as part of the input that must be preserved by the drawing algorithm ...
A drawing of a graph is a monotone drawing if for every pair of vertices u and v, there is a path drawn from u to v that is monotone in some direction. ...
this section we study monotone drawings of embedded planar graphs. ...
doi:10.1007/978-3-642-25878-7_36
fatcat:d6ofhrdr6bex5opoir45xg2nue
Monotone Drawings of Graphs with Fixed Embedding
2013
Algorithmica
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In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as part of the input that must be preserved by the drawing algorithm ...
A drawing of a graph is a monotone drawing if for every pair of vertices u and v, there is a path drawn from u to v that is monotone in some direction. ...
this section we study monotone drawings of embedded planar graphs. ...
doi:10.1007/s00453-013-9790-3
fatcat:lbb3ofk56ncy3ct6kclavn4e74
On the number of upward planar orientations of maximal planar graphs
2014
Theoretical Computer Science
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We consider the problem of determining the maximum and the minimum number of upward planar orientations a maximal planar graph can have. ...
Moreover, there exist n-vertex maximal planar graphs having as few as O(n · 2 n ) upward planar orientations and n-vertex maximal planar graphs having Ω(2.599 n ) upward planar orientations. ...
upward planar drawings of directed graphs (see, e.g., [1, 6] ). ...
doi:10.1016/j.tcs.2014.04.024
fatcat:gkwwxhkj4vgllne4uukg7u4vxy
On the Number of Upward Planar Orientations of Maximal Planar Graphs
[chapter]
2012
Lecture Notes in Computer Science
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We consider the problem of determining the maximum and the minimum number of upward planar orientations a maximal planar graph can have. ...
Moreover, there exist n-vertex maximal planar graphs having as few as O(n · 2 n ) upward planar orientations and n-vertex maximal planar graphs having Ω(2.599 n ) upward planar orientations. ...
upward planar drawings of directed graphs (see, e.g., [1, 6] ). ...
doi:10.1007/978-3-642-35261-4_44
fatcat:yxunzi5uizbnhehk7sktc7mhy4
Straightening out planar poly-line drawings
[article]
2014
arXiv
pre-print
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Likewise y-monotonicity is required: there are poly-line drawings (not y-monotone) that cannot be straightened out while maintaining the height. We give some applications of our result. ...
We show that any y-monotone poly-line drawing can be straightened out while maintaining y-coordinates and height. ...
Let Γ be a planar y-monotone poly-line drawing of a triangulated graph G. ...
arXiv:1404.5892v2
fatcat:aumv4cy43zbjbbjwimhqf2dali
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