## Simultaneous Embedding of a Planar Graph and Its Dual on the Grid (2002)

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Venue: | In 13th Intl. Symp. on Algorithms and Computation (ISAAC |

Citations: | 14 - 9 self |

### BibTeX

@INPROCEEDINGS{Erten02simultaneousembedding,

author = {Cesim Erten and Stephen G. Kobourov},

title = {Simultaneous Embedding of a Planar Graph and Its Dual on the Grid},

booktitle = {In 13th Intl. Symp. on Algorithms and Computation (ISAAC},

year = {2002},

pages = {575--587},

publisher = {SpringerVerlag}

}

### Years of Citing Articles

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### Abstract

Traditional representations of graphs and their duals suggest the requirement that the dual vertices should be placed inside their corresponding primal faces, and the edges of the dual graph should cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual on a small integer grid such that the edges are drawn as straight-line segments and the only crossings are between primal-dual pairs of edges. We provide an O(n) time algorithm that simultaneously embeds a 3-connected planar graph anditsdualona(2n-2) 2) integer grid, where n is the total number of vertices in the graph and its dual.

### Citations

674 | LEDA: A Platform for Combinatorial and Geometric Computing
- Mehlhorn, Naher
- 1999
(Show Context)
Citation Context ...otal area required is (2n − 2) × (2n − 2) and Theorem-1 follows. 3 Implementation We have implemented our algorithm to visualize 3-connected planar graphs and their duals using the LEDA/AGD libraries =-=[10]-=-. Finding a suitable canonical labeling takes linear time [9]. We make use of the technique introduced by [5] to do the placement step. It is based on the fact that storing relative x-coordinates of t... |

395 | How to draw a graph
- Tutte
- 1963
(Show Context)
Citation Context ...ected planar graphs with at least 4 vertices on the outer face and present an algorithm for straight-line embedding of such graphs on a (⌈n/2⌉−1) × (⌊n/2⌋) grid. In a paper dating back to 1963, Tutte =-=[14]-=- shows that there exists a simultaneous straight-line representation of any planar graph and its dual in which the only intersections are between corresponding primal-dual edges. However, a disadvanta... |

238 | Efficient planarity testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...lgorithm is O(n). 2.1 Overview of the Algorithm Given a 3-connected graph G1, we summarize our algorithm to simultaneously embed G1 and its dual as follows: • Find a topological embedding of G1 using =-=[8]-=-. • Apply the construction described above to find G2. • Let G = G2, whereGis an FQ-3-connected planar graph. • Find a suitable canonical labeling of the vertices of G. • Place the vertices of G on th... |

199 |
Embedding planar graphs on the grid
- Schnyder
- 1990
(Show Context)
Citation Context ...tended abstract is at www.cs.arizona.edu/~cesim/dual.ps.Pach and Pollack [6] who provide an algorithm that embeds a planar graph on n vertices on the (2n − 4) × (n − 2) integer grid. Later, Schnyder =-=[13]-=- present another method that requires grid size (n−2)×(n−2). Also, several restrictions of this problem have been considered. Harel and Sardas [7] provide an algorithm to embed a biconnected graph on ... |

156 |
How to draw a planar graph on a grid
- Fraysseix, Pach, et al.
- 1990
(Show Context)
Citation Context ... well-studied graph drawing problem. The first solution to this problem was given by de Fraysseix, ⋆ A full version of this extended abstract is at www.cs.arizona.edu/~cesim/dual.ps.Pach and Pollack =-=[6]-=- who provide an algorithm that embeds a planar graph on n vertices on the (2n − 4) × (n − 2) integer grid. Later, Schnyder [13] present another method that requires grid size (n−2)×(n−2). Also, severa... |

129 |
On straight line representation of planar graphs
- FÁRY
- 1948
(Show Context)
Citation Context .... ? Partially supported by NSF grant ACR-0222920. 1 1.1 Related Work While Fary proved the existence of crossings-free straight-line drawings for planar graphs (also known as Fary drawings) in 1948 =-=[8]-=-, it was not until 1963 that thesrst algorithm for producing such drawings was introduced by Tutte [17]. In addition to describing a method for producing Fary drawings of planar graphs, Tutte also sh... |

69 |
Rectilinear planar layouts and bipolar orientations of planar graphs
- Rosenstiehl, Tarjan
- 1986
(Show Context)
Citation Context ... locations of the vertices, however, requires high-precision operations and results in drawings with uneven distribution of the vertices of the drawing area. With this in mind, Rosenstiehl and Tarjan =-=[14]-=- observed that when drawing planar graphs, it is desirable to map the vertices of the graph on a small integer grid. Such integer grid drawings have good distribution of the vertices over the drawing ... |

68 | Drawing planar graphs using the canonical ordering
- Kant
- 1996
(Show Context)
Citation Context ...r we consider only 3-connected graphs. 2.2 The Canonical Labeling We present the canonical labeling for the type of graphs under consideration. It is a simple restriction of the canonical labeling of =-=[9]-=-, which in turn is based on the ordering defined in [6]. Let G be an FQ-3-connected planar graph with n vertices. Let (u, v, w, w ′ ) be the outer face of G s.t. u, w are primal vertices and v, w ′ ar... |

53 |
Planar graphs and poset dimension
- Schnyder
- 1989
(Show Context)
Citation Context ...was described by de Fraysseix, Pach and Pollack [7]. The method relies on computing a shelling order of the vertices and carefully inserting them on the grid using that order. Independently, Schnyder =-=[15, 16]-=- describes a method based on barycentric coordinates that requires grid size (n2)(n2). Restrictions of this problem have been considered as well. Harel and Sardas [9] provide an algorithm to embed ... |

46 |
Small sets supporting Fáry embeddings of planar graphs
- Fraysseix, Pach, et al.
- 1988
(Show Context)
Citation Context ...e drawn as a straight-line segment and that no crossings between edges are created, is a wellstudied graph drawing problem. The first solution to this problem was given by Fraysseix, Pach and Pollack =-=[5]-=- who provided an algorithm that embeds a planar graph on n vertices on the (2n − 4) × (n − 2) integer grid. Later, Schnyder [10] developed another method that reduces the grid size to (n − 2) × (n − 2... |

37 | Convex grid drawings of 3-connected planar graphs
- Chrobak, Kant
(Show Context)
Citation Context ...blem have been considered. Harel and Sardas [7] provide an algorithm to embed a biconnected graph on the (2n−4)×(n−2) grid without triangulating the graph initially. The algorithm of Chrobak and Kant =-=[4]-=- embeds a 3-connected planar graph on a (n−2)×(n−2) grid so that each face is convex. Miura, Nakano, and Nishizeki [11] further restrict the graphs under consideration to 4-connected planar graphs wit... |

37 | A linear-time algorithm for drawing planar graphs
- Chrobak, Payne
- 1995
(Show Context)
Citation Context ...rithm to visualize 3-connected planar graphs and their duals using the LEDA/AGD libraries [10]. Finding a suitable canonical labeling takes linear time [9]. We make use of the technique introduced by =-=[5]-=- to do the placement step. It is based on the fact that storing relative x-coordinates of the previously embedded vertices is sufficient at every step. Then the placement step also requires only linea... |

31 | Convex drawings of graphs in two and three dimensions
- CHROBAK, GOODRICH, et al.
- 1996
(Show Context)
Citation Context ...n the grid so that each internal face of G is strictly convex and the outer face of G lies on a strictly concave quadrilateral. Note that this problem can be solved by the algorithm of Chrobak et al. =-=[3]-=-. However, the area guaranteed by their algorithm is O(n 3 ) × O(n 3 ), whereas our algorithm guarantees a drawing on the (2n − 2) × (2n − 2) grid, which is stated in the main theorem in this paper: T... |

30 |
Representations of planar graphs
- Brightwell, Scheinerman
- 1993
(Show Context)
Citation Context ...ding primal-dual edges. However, a disadvantage of this representation is that the area required by the algorithm can be exponential in the number of vertices of the graph. Brightwell and Scheinerman =-=[2]-=- show that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and each face, so that, for each edge of G, the four circles representing... |

27 |
E#cient planarity testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...lgorithm is O(n). 2.1 Overview of the Algorithm Given a 3-connected graph G1, we summarize our algorithm to simultaneously embed G1 and its dual as follows: Find a topological embedding of G1 using =-=[10]-=-. Apply the construction described above tosnd G2. Let G = G2, where G is a maximal bipartite 3-connected planar graph. Find a suitable canonical labeling of the vertices of G. Place the verti... |

16 |
On simultaneous graph embedding
- Brass, Cenek, et al.
- 2003
(Show Context)
Citation Context ...ded faces and show that the problem is NP-hard for the case of convex 5-sided faces. Another related problem is that of simultaneously embedding more than one planar graph. In particular, Brass et al =-=[2]-=- consider the problem of simultaneous embedding of pairs of planar graphs. Given two planar graphs on the same set of vertices, H1 = (V; E1) and H2 = (V; E2), the goal is to embed H1 and H2 simultaneo... |

13 | An algorithm for straight-line drawing of planar graphs
- Harel, Sardas
- 1998
(Show Context)
Citation Context ...the (2n − 4) × (n − 2) integer grid. Later, Schnyder [13] present another method that requires grid size (n−2)×(n−2). Also, several restrictions of this problem have been considered. Harel and Sardas =-=[7]-=- provide an algorithm to embed a biconnected graph on the (2n−4)×(n−2) grid without triangulating the graph initially. The algorithm of Chrobak and Kant [4] embeds a 3-connected planar graph on a (n−2... |

9 | Circle packing of maps in polynomial time
- Mohar
- 1997
(Show Context)
Citation Context ...eously in the plane with straight-line edges so that the primal edges cross the dual edges at right angles (provided that the vertex corresponding to the unbounded face is located at infinity). Mohar =-=[12]-=- extends the results of [2] by presenting an approximation algorithm that given a 3-connected planar graph G = (V,E) and a rational number ɛ>0 finds an ɛ-approximation for the radii and the coordinate... |

8 |
Drawing the planar dual
- Bern, Gilbert
- 1992
(Show Context)
Citation Context ...cle representation for G and its dual. Mohar’s algorithm runs in time polynomial in |E(G)| and log(1/ɛ) and the angles of the primal-dual edge crossings are arbitrarily close to π/2. Bern and Gilb=-=ert [1]-=- address a variation of the simultaneous planar-dual embedding problem: finding suitable locations for dual vertices, given a straightline planar embedding of a planar graph, so that the edges of the ... |

4 |
Grid drawings of 4-connected plane graphs
- Miura, Nakano, et al.
(Show Context)
Citation Context ...grid without triangulating the graph initially. The algorithm of Chrobak and Kant [4] embeds a 3-connected planar graph on a (n−2)×(n−2) grid so that each face is convex. Miura, Nakano, and Nishizeki =-=[11]-=- further restrict the graphs under consideration to 4-connected planar graphs with at least 4 vertices on the outer face and present an algorithm for straight-line embedding of such graphs on a (⌈n/2⌉... |