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### Grid Drawings of Graphs with Constant Edge-Vertex Resolution [article]

Michael A. Bekos, Martin Gronemann, Fabrizio Montecchiani, Dömötör Pálvölgyi, Antonios Symvonis, Leonidas Theocharous
2021 arXiv   pre-print
We call such drawings disk-link drawings. In this paper we focus on the case of constant edge-vertex resolution, namely ρ=1/2 (i.e., disks of unit diameter).  ...  We study the algorithmic problem of computing drawings of graphs in which (i) each vertex is a disk with fixed radius ρ, (ii) each edge is a straight-line segment connecting the centers of the two disks  ...  In relation to our work, they state that every 3-connected planar graph with n vertices admits a convex grid drawing with constant edge-vertex resolution in (3n − 7) × (3n − 7)/2 area.  ...

### Convex Grid Drawings of Planar Graphs with Constant Edge-Vertex Resolution [article]

Michael A. Bekos, Martin Gronemann, Fabrizio Montecchiani, Antonios Symvonis
2022 arXiv   pre-print
We present a drawing algorithm that takes as input a 3-connected plane graph with n vertices and f internal faces and computes a convex straight-line drawing with edge-vertex resolution at least 1/2 on  ...  edge is lower bounded by a constant that does not depend on the size of the graph.  ...  A disk-link drawing of a graph is a grid drawing of edge-vertex resolution at least 1 2 .  ...

### Limitations on Realistic Hyperbolic Graph Drawing [article]

David Eppstein
2021 arXiv   pre-print
We show that several types of graph drawing in the hyperbolic plane require features of the drawing to be separated from each other by sub-constant distances, distances so small that they can be accurately  ...  Therefore, for these types of drawing, hyperbolic geometry provides no benefit over Euclidean graph drawing.  ...  Grid graphs also obey similar bounds (Theorem 5). -Beyond planar graph drawing, every n-vertex graph has a Euclidean drawing with unit vertex-vertex resolution and angular resolution Θ(1/n).  ...

### Convex drawings of graphs in two and three dimensions (preliminary version)

Marek Chrobak, Michael T. Goodrich, Roberto Tamassia
1996 Proceedings of the twelfth annual symposium on Computational geometry - SCG '96
We provide O(n)-time algorithms for constructing the following types of drawings of n-vertex 3-connected planar graphs: 2D convex grid drawings with (3n) (3n=2) area under the edge L 1 -resolution rule  ...  ; 2D strictly convex grid drawings with O(n 3 ) O(n 3 ) area under the edge resolution rule; 2D strictly convex drawings with O(1) O(n) area under the vertex-resolution rule, and with vertex coordinates  ...  Abstract We provide O(n)-time algorithms for constructing the following types of drawings of n-vertex 3-connected planar graphs: 2D convex grid drawings with (3n) (3n=2) area under the edge L We also  ...

### Drawing Planar Graphs with Circular Arcs [chapter]

C. C. Cheng, C. A. Duncan, M. T. Goodrich, S. G. Kobourov
1999 Lecture Notes in Computer Science
In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area.  ...  The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining Θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v.  ...  -We can draw an n-vertex planar graph G in an O(n)×O(n) grid with angular resolution Θ (1/d(v) ) for each vertex v in G using at most two circular arcs per edge.  ...

### Graph drawing [chapter]

Roberto Tamassia, Giuseppe Liotta
2004 Handbook of Discrete and Computational Geometry, Second Edition
resolution ( ): smallest angle formed by two edges, or segments of edges, incident on the same vertex or bend, in a polyline drawing.  ...  maxi- mum angular resolution NP-hard 55, 73] embedded graph test the existence of a planar straight-line drawing with prescribed angles between pairs of consecutive edges incident on a vertex NP-hard  ...

### Optimal 3D Angular Resolution for Low-Degree Graphs

David Eppstein, Maarten Löffler, Elena Mumford, Martin Nöllenburg
2013 Journal of Graph Algorithms and Applications
We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5-degree angles, i.e., the angular resolution of the diamond lattice, between  ...  We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend  ...  This research was supported in part by the National Science Foundation under grant 0830403, by the Office of Naval Research under MURI grant N00014-08-1-1015, and by the German Research Foundation (DFG  ...

### Optimal 3D Angular Resolution for Low-Degree Graphs [chapter]

David Eppstein, Maarten Löffler, Elena Mumford, Martin Nöllenburg
2011 Lecture Notes in Computer Science
We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5 • angles, i. e., the angular resolution of the diamond lattice, between  ...  We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120 • angles between any two edge segments meeting at a vertex or a bend.  ...  This research was supported in part by the National Science Foundation under grant 0830403, by the Office of Naval Research under MURI grant N00014-08-1-1015, and by the German Research Foundation (DFG  ...

### Advances in the theory and practice of graph drawing

Roberto Tamassia
1999 Theoretical Computer Science
Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph drawing.  ...  The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering.  ...  Some columns of the grid are "reserved" to draw the remaining incident edges of v.  ...

### Trade-Offs in Planar Polyline Drawings [chapter]

Stephane Durocher, Debajyoti Mondal
2014 Lecture Notes in Computer Science
We present the first smooth tradeoff between the area and angular resolution for 2-bend polyline drawings of any given planar graph.  ...  Specifically, for any given n-vertex triangulation, our algorithm computes a drawing with angular resolution r/d(v) at each vertex v, and area f (n, r), for any r ∈ (0, 1], where d(v) denotes the degree  ...  Allowing bends helps both to reduce area and to improve angular resolution, e.g., given an n-vertex planar graph with maximum degree ∆, one can construct a 3-bend polyline drawing with 2/∆ radians of resolution  ...

### Area, Curve Complexity, and Crossing Resolution of Non-planar Graph Drawings [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer
2010 Lecture Notes in Computer Science
In this paper we study non-planar drawings of graphs, and study tradeoffs between the crossing resolution (i.e., the minimum angle formed by two crossing segments), the curve complexity (i.e., maximum  ...  number of bends per edge), the total number of bends, and the area.  ...  It would be interesting to study whether a o(n 2 ) area and good crossing resolution can be obtained for graphs with O(n) edges.  ...

### Small Area Drawings of Outerplanar Graphs [chapter]

Giuseppe Di Battista, Fabrizio Frati
2006 Lecture Notes in Computer Science
On the other hand, the first algorithm constructs drawings with better angular resolution. The third algorithm constructs outerplanar drawings of general outerplanar graphs with O(n 1.48 ) area.  ...  We show three linear time algorithms for constructing planar straightline grid drawings of outerplanar graphs. The first and the second algorithm are for balanced outerplanar graphs.  ...  We show (Section 3) a linear time algorithm for constructing nonouterplanar drawings of balanced outerplanar graphs in linear area and with angular resolution ≥ c √ n , with c constant.  ...

### A Framework for Drawing Planar Graphs with Curves and Polylines

Michael T. Goodrich, Christopher G. Wagner
2000 Journal of Algorithms
This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex  ...  More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) × O(n) integer grid so that  ...  One of the now-classic results for drawing planar graphs is an algorithm by de Fraysseix, Pach, and Pollack  for drawing an n-vertex planar graph in an O(n) × O(n) integer grid using straight line  ...

### A Framework for Drawing Planar Graphs with Curves and Polylines [chapter]

Michael T. Goodrich, Christopher G. Wagner
1998 Lecture Notes in Computer Science
This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex  ...  More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) × O(n) integer grid so that  ...  One of the now-classic results for drawing planar graphs is an algorithm by de Fraysseix, Pach, and Pollack  for drawing an n-vertex planar graph in an O(n) × O(n) integer grid using straight line  ...

### Polar Coordinate Drawing of Planar Graphs with Good Angular Resolution

Christian Duncan, Stephen Kobourov
2003 Journal of Graph Algorithms and Applications
In both algorithms we are concerned with the following drawing criteria: angular resolution, bends per edge, vertex resolution, bend-point resolution, edge separation, and drawing area.  ...  We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms for constructing it.  ...  Acknowledgments A preliminary version of this paper appeared in the Proceedings of the 9th Symposium on Graph Drawing  .  ...
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