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### Monotone Drawings of Planar Graphs [chapter]

János Pach, Gézá Tóth
2002 Lecture Notes in Computer Science
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.  ...

### Monotone drawings of planar graphs [article]

Janos Pach, Geza Toth
2011 arXiv   pre-print
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.  ...

### Monotone drawings of planar graphs

János Pach, Géza Tóth
2004 Journal of Graph Theory
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.  ...

### Monotone Drawings of k-Inner Planar Graphs [article]

Anargyros Oikonomou, Antonios Symvonis
2018 arXiv   pre-print
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  ...

### Monotone Grid Drawings of Planar Graphs [chapter]

Md. Iqbal Hossain, Md. Saidur Rahman
2014 Lecture Notes in Computer Science
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.  ...

### Strongly Monotone Drawings of Planar Graphs [article]

Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, Manfred Scheucher
2016 arXiv   pre-print
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.  ...

### Monotone Grid Drawings of Planar Graphs [article]

Md. Iqbal Hossain, Md. Saidur Rahman
2013 arXiv   pre-print
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.  ...

### Strongly Monotone Drawings of Planar Graphs

Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, Manfred Scheucher, Marc Herbstritt
2016 International Symposium on Computational Geometry
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.  ...

### Strongly Monotone Drawings of Planar Graphs *

Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, Manfred Scheucher
unpublished
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.  ...

### Height-Preserving Transformations of Planar Graph Drawings [chapter]

Therese Biedl
2014 Lecture Notes in Computer Science
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.  ...

### Monotone Drawings of Graphs with Fixed Embedding [chapter]

Patrizio Angelini, Walter Didimo, Stephen Kobourov, Tamara Mchedlidze, Vincenzo Roselli, Antonios Symvonis, Stephen Wismath
2012 Lecture Notes in Computer Science
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.  ...

### Monotone Drawings of Graphs with Fixed Embedding

Patrizio Angelini, Walter Didimo, Stephen Kobourov, Tamara Mchedlidze, Vincenzo Roselli, Antonios Symvonis, Stephen Wismath
2013 Algorithmica
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.  ...

### On the number of upward planar orientations of maximal planar graphs

Fabrizio Frati, Joachim Gudmundsson, Emo Welzl
2014 Theoretical Computer Science
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] ).  ...

### On the Number of Upward Planar Orientations of Maximal Planar Graphs [chapter]

Fabrizio Frati, Joachim Gudmundsson, Emo Welzl
2012 Lecture Notes in Computer Science
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] ).  ...

### Straightening out planar poly-line drawings [article]

Therese Biedl
2014 arXiv   pre-print
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.  ...
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