## Drawing Planar Graphs Using the Canonical Ordering (1996)

Venue: | ALGORITHMICA |

Citations: | 68 - 0 self |

### BibTeX

@ARTICLE{Kant96drawingplanar,

author = {Goos Kant},

title = {Drawing Planar Graphs Using the Canonical Ordering},

journal = {ALGORITHMICA},

year = {1996},

volume = {16},

pages = {4--32}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems. -- Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices. -- Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends. -- Every 3-planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid. -- Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...

### Citations

544 | A framework for dynamic graph drawing
- Cohen, Battista, et al.
- 1992
(Show Context)
Citation Context ...ar with straight lines such that the minimum angle is ? \Omega\Gamma 1 d )? (See also [31] and [12] for this question.) ffl Devise a dynamic algorithms for the sequential algorithms of this paper. In =-=[6]-=- a dynamic framework for graph drawing problems is described, but this approach seems not to work here. Very recently, He & Kao presented a parallel implementation for finding a canonical ordering, an... |

485 | Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-tree Algorithms - Booth, Lueker - 1976 |

238 | Efficient planarity testing - Hopcroft, Tarjan - 1974 |

185 | Dividing a graph into triconnected components - Hopcroft, Tarjan - 1973 |

129 | On straight line representation of planar graphs - FÁRY - 1948 |

112 | A linear algorithm for embedding planar graphs using PQ-trees - Chiba, Nishizeki, et al. - 1985 |

86 | On the computational complexity of upward and rectilinear planarity testing - Garg, Tamassia - 1995 |

82 | Incremental Planarity Testing
- Battista, Tamassia
- 1989
(Show Context)
Citation Context ...es (polygons). The triconnected components are unique. Next we need the SPQR-tree, a versatile data structure that represents the decomposition of a biconnected graph into its triconnected components =-=[10]-=-. The SPQR-tree T is defined as follows: for every triconnected component we create an R-node, for every polygon an S-node, for every bond a P-node, and for every edge a Q-node. The edges in T are def... |

64 |
How to draw a planar graph on a grid, Combinatorica 10
- Fraysseix, Pach, et al.
- 1990
(Show Context)
Citation Context ... It is well-known that every planar graph can be drawn planar with straight lines, and by more recent algorithms this can be done in linear time and space on a grid of size O(n) \Theta O(n) (see e.g. =-=[7, 17, 19, 34]-=-).\Omega\Gamma n 2 ) is also a lower bound for the area of planar straight-line drawings [17]. However, a drawback of all these drawing algorithms is that the minimum angle between lines can be very s... |

37 | Convex grid drawings of 3-connected planar graphs
- Chrobak, Kant
(Show Context)
Citation Context ...ces, where the edges contain the amount of shift for every edge. When applying this techique for our canonical ordering, there is no reason to compute a leftmost canonical ordering. In Chrobak & Kant =-=[8]-=-, this is described in more detail. Since every lmc-ordering is also an st-ordering, we can use the lmc-ordering in various drawing applications, where the st-ordering is used. We now focus the attent... |

35 |
Linear algorithms for convex drawings of planar graphs
- Chiba, Yamanouchi, et al.
- 1984
(Show Context)
Citation Context ... be drawn convexly. Tutte showed that every triconnected planar graph can be drawn with convex faces. Thomassen characterized the class of planar graphs which admit a convex drawing, and Chiba et al. =-=[5]-=- presented a linear time drawing algorithm for this class. However, the coordinates of the vertices can be reals and a huge number of vertices can be clustered in a small area. Another representation ... |

31 |
Algorithms for automatic graph drawing: An annotated bibliography
- Eades, Tamassia
- 1989
(Show Context)
Citation Context ...nds, or satisfies some other constraint. See the annotated bibliography for an up to date overview of the recent developments and optimization problems in graph drawings with more than 300 references =-=[9]-=-. It is well-known that every planar graph can be drawn planar with straight lines, and by more recent algorithms this can be done in linear time and space on a grid of size O(n) \Theta O(n) (see e.g.... |

30 |
A Linear Algorithm to Find a Rectangular Dual of a Planar Triangulated Graph
- Bhasker, Sahni
- 1988
(Show Context)
Citation Context ...ne bend (if n ? 4). We notice that better bounds can be obtained if the dual graph G of G is a 4-connected planar graph in which all internal faces are triangles. It has been shown by Bhasker & Sahni =-=[1]-=- that in this case G can be drawn orthogonally in linear time such that there are at most 4 bends. 5.3 General 3-Planar Graphs To apply the algorithm to general connected 3-planar graphs, we have to s... |

30 | Drawing graphs in the plane with high resolution - Formann, Hagerup, et al. - 1993 |

22 | On finding the rectangular duals of planar triangular graphs - He - 1993 |

20 |
A linear time algorithm for drawing a planar graph on a grid
- Chrobak, Payne
- 1990
(Show Context)
Citation Context ... It is well-known that every planar graph can be drawn planar with straight lines, and by more recent algorithms this can be done in linear time and space on a grid of size O(n) \Theta O(n) (see e.g. =-=[7, 17, 19, 34]-=-).\Omega\Gamma n 2 ) is also a lower bound for the area of planar straight-line drawings [17]. However, a drawback of all these drawing algorithms is that the minimum angle between lines can be very s... |

20 |
Rectilinear planar drawings with few bends in each edge
- Even, Granot
- 1994
(Show Context)
Citation Context ...ge with three bends? Indeed, Even & Granot proved that any orthogonal drawing of the 4-planar triangulated planar graph on 6 vertices (octahedron) requires at least one edge with at least three bends =-=[13]-=-. In our case, if there is a vertex v with deg(v) = 3 then we can set v n = v. Otherwise let n ? 6. Then there is a face with at least 4 vertices, which we choose to be the outerface. Let v n 1 ; : : ... |

9 | Area requirement and symmetry display in drawing graphs - DiBattista, Tamassia, et al. - 1989 |

9 | Corrigendum: “Computing an st-numbering” (Theoret - Even, Tarjan - 1976 |

4 | Planar embedding: Linear-time algorithms for vertex placement and edge ordering - Jayakumar, Thulasiraman, et al. - 1988 |

1 |
A Better Heuristic for Planar Orthogonal Drawings
- Biedl, Kant
- 1994
(Show Context)
Citation Context ... n ), resp.; End 4-Orthogonal See Figure 6 for an illustration of the different cases. There are several ways for computing the coordinates. Here we briefly describe the method, given by Biedl & Kant =-=[2]-=-: Remark that the y-coordinate of a vertex is never changed later, so we only have to worry about the x-coordinates. The crucial observation is that we need not know the values of the x-coordinates of... |

1 |
Angles of Planar Triangular Graphs, extended abstract in
- Battista, Vismara
- 1993
(Show Context)
Citation Context ... embedded 4-planar graph on an n \Theta n grid with the minimum number of bends [36]. If the planar embedding is not given in advance, then the problem is polynomial time solvable for 3-planar graphs =-=[12]-=-, and NP-hard for 4-planar graphs [18]. In particular, Garg & Tamassia showed that is even NP-hard to approximate the minimum number of bends in a planar orthogonal drawing with an O(n 1\Gammaffl ) er... |

1 |
Straight Line Embeddings on the Grid
- Haandel
- 1991
(Show Context)
Citation Context ... It is well-known that every planar graph can be drawn planar with straight lines, and by more recent algorithms this can be done in linear time and space on a grid of size O(n) \Theta O(n) (see e.g. =-=[7, 17, 19, 34]-=-).\Omega\Gamma n 2 ) is also a lower bound for the area of planar straight-line drawings [17]. However, a drawback of all these drawing algorithms is that the minimum angle between lines can be very s... |

1 | Hexagonal Grid Drawings, in: E.W - Kant |