## Drawings of planar graphs with few slopes and segments (2005)

Venue: | Computational Geometry Theory and Applications 38:194–212 |

Citations: | 15 - 4 self |

### BibTeX

@INPROCEEDINGS{Dujmović05drawingsof,

author = {Vida Dujmović and Matthew Suderman and David R. Wood},

title = {Drawings of planar graphs with few slopes and segments},

booktitle = {Computational Geometry Theory and Applications 38:194–212},

year = {2005},

pages = {122--132},

publisher = {Springer}

}

### OpenURL

### Abstract

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5 2n segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.

### Citations

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(Show Context)
Citation Context ...10 slopes. most 5 2 The proof of Theorem 6 is based on the canonical ordering of Kant [16], which is a generalisation of a similar structure for plane triangulations introduced by de Fraysseix et al. =-=[8]-=-. Let G be a 3-connected plane graph. Kant [16] proved that G has a canonical ordering defined as follows. Let σ = (V1, V2, . . . , VK) be an ordered partition of V (G). That is, V1 ∪ V2 ∪ · · · ∪ VK ... |

123 |
On straight line representation of planar graphs
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(Show Context)
Citation Context .... We emphasise that a plane drawing of a plane graph must preserve the embedding and outerface. That every plane graph has a plane drawing is a famous result independently due to Wagner [26] and Fáry =-=[12]-=-. In this paper we prove lower and upper bounds on the minimum number of segments and slopes in (plane) drawings of graphs. In a companion paper [10], we consider drawings of non-planar graphs with fe... |

114 |
On Rigid Circuit Graphs
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Citation Context ...ose that the claim holds for plane 3-trees on n − 1 vertices. Let G be a plane 3-tree on n vertices. Every k-tree on at least k+2 vertices has two non-adjacent simplicial vertices of degree exactly k =-=[9]-=-. In particular, G has two non-adjacent simplicial degree-3 vertices, one of which, say v, is not on the outerface. Thus G can be obtained from G\v by adding v inside some internal face (p, q, r) of G... |

96 |
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Citation Context ...fied outerface. We emphasise that a plane drawing of a plane graph must preserve the embedding and outerface. That every plane graph has a plane drawing is a famous result independently due to Wagner =-=[26]-=- and Fáry [12]. In this paper we prove lower and upper bounds on the minimum number of segments and slopes in (plane) drawings of graphs. In a companion paper [10], we consider drawings of non-planar ... |

81 | On the computational complexity of upward and rectilinear planarity testing. Graph Drawing
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- 1995
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Citation Context ...nly if they cross in D ′ . Thus D ′ is plane whenever D is plane. Corollary 1. A graph has a (plane) drawing on two slopes if and only if it has a (plane) drawing on any two slopes. Garg and Tamassia =-=[13]-=- proved that it is N P-complete to decide whether a graph has a rectilinear planar drawing (that is, with vertical and horizontal edges). Thus Corollary 1 implies: Corollary 2. It is N P-complete to d... |

65 | Drawing planar graphs using the canonical ordering
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(Show Context)
Citation Context ...orem 6. Every 3-connected plane graph with n vertices has a plane drawing with at n − 3 segments and at most 2n − 10 slopes. most 5 2 The proof of Theorem 6 is based on the canonical ordering of Kant =-=[16]-=-, which is a generalisation of a similar structure for plane triangulations introduced by de Fraysseix et al. [8]. Let G be a 3-connected plane graph. Kant [16] proved that G has a canonical ordering ... |

29 | A linear algorithm to find a rectangular dual of a planar triangulated graph - Bhasker, Sahni - 1988 |

20 | A left-first search algorithm for planar graphs. Discrete Computational Geometry 13:459–468
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Citation Context ... horizontal and vertical segments. A contact graph is an intersection graph of segments for which no two segments have an interior point in common. Strengthening the above result, de Fraysseix et al. =-=[7]-=- (and later, Czyzowicz et al. [4]) proved that every bipartite planar graph is a contact graph of some set of horizontal and vertical segments. Similarly, de Castro et al. [5] proved that every triang... |

17 | Triangle-free planar graphs and segment intersection graphs
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(Show Context)
Citation Context ...ult, de Fraysseix et al. [7] (and later, Czyzowicz et al. [4]) proved that every bipartite planar graph is a contact graph of some set of horizontal and vertical segments. Similarly, de Castro et al. =-=[5]-=- proved that every triangle-free planar graph is a contact graph of some set of segments with only three distinct slopes. It is an open problem whether every planar graph is the intersection graph of ... |

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14 |
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Citation Context ...rch. The intersection graph of a set of segments has one vertex for each segment, and two vertices are adjacent if and only if the corresponding segments have a non-empty intersection. Hartman et al. =-=[14]-=- proved that every bipartite planar graph is the intersection graph of some set of horizontal and vertical segments. A contact graph is an intersection graph of segments for which no two segments have... |

13 | Bounded-degree graphs have arbitrarily large geometric thickness
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(Show Context)
Citation Context ...plane drawing with f(∆) slopes? This is open even for maximal outerplanar graphs. Note that there exist bounded degree (non-planar) graphs for which the number of slopes is unbounded in every drawing =-=[1, 10, 18]-=-. The best bounds are in our companion paper [10], in which we prove that there exists ∆-regular n-vertex graphs with at least n 1− 8+ɛ ∆+4 slopes in every drawing. Open Problem 5. In all our results,... |

13 | Graph treewidth and geometric thickness parameters
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(Show Context)
Citation Context ...osition a set of vertices at each step, rather than a single vertex. The next lemma says how to partition a 2-tree appropriately. It has subsequently been generalised for k-trees by Dujmović and Wood =-=[11]-=-. Lemma 4. Let G be a 2-tree. Then for some k ≥ 1, V (G) can be partitioned (S0, S1, S2, . . . , Sk) such that: (a) for 1 ≤ i ≤ k, the subgraph Gi = G[ � i j=0 Sj] is a 2-tree, (b) S0 consists of two ... |

10 | Hexagonal grid drawings
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(Show Context)
Citation Context ...imal for any 3-connected cubic plane graph whose outerface is a triangle. It is easily seen that there is such a graph on n vertices for all even n ≥ 4. • Theorem 8 was independently obtained by Kant =-=[15]-=-. We believe that our proof is much simpler. Kant [15] also claimed to prove that every plane graph with maximum degree three has a 3-slope drawing (except for one bent edge). This claim is false. Con... |

10 |
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(Show Context)
Citation Context ... bends on the outerface. We now review some related work from the literature. Plane orthogonal drawings with two slopes (and few bends) have been extensively studied [2, 3, 19–25]. For example, Ungar =-=[25]-=- proved that every cyclically 4-edge-connected plane cubic graph has a plane drawing with two slopes and four bends on the outerface. Thus our result for 3-connected plane cubic graphs (Corollary 4) n... |

10 |
de Fraysseix, Patrice Ossona de Mendez, and János Pach. A leftfirst search algorithm for planar graphs
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(Show Context)
Citation Context ...nar graph is a contact graph of some set of segments with only three distinct slopes. It is an open problem whether every planar graph is the intersection graph of a set of segments in the plane; see =-=[6, 17]-=- for the most recent results. It is even possible that every k-colourable planar graph (k ≤ 4) is the intersection graph of some set of segments using only k distinct slopes. 3s1.1 Definitions We cons... |

7 | A linear time algorithm to check for the existence of a rectangular dual of a planar triangulated graph - Bhasker, Sahni - 1987 |

6 |
A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments
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(Show Context)
Citation Context .... A contact graph is an intersection graph of segments for which no two segments have an interior point in common. Strengthening the above result, de Fraysseix et al. [7] (and later, Czyzowicz et al. =-=[4]-=-) proved that every bipartite planar graph is a contact graph of some set of horizontal and vertical segments. Similarly, de Castro et al. [5] proved that every triangle-free planar graph is a contact... |

5 | Graph drawings with few slopes
- DUJMOVIĆ, SUDERMAN, et al.
(Show Context)
Citation Context ...s result independently due to Wagner [26] and Fáry [12]. In this paper we prove lower and upper bounds on the minimum number of segments and slopes in (plane) drawings of graphs. In a companion paper =-=[10]-=-, we consider drawings of non-planar graphs with few slopes. A summary of our results is given in Table 1. A number of comments are in order when considering these results: • The minimum number of slo... |

3 |
and Dömötör Pálvölgyi. Bounded-degree graphs can have arbitrarily large slope numbers
- Pach
(Show Context)
Citation Context ...plane drawing with f(∆) slopes? This is open even for maximal outerplanar graphs. Note that there exist bounded degree (non-planar) graphs for which the number of slopes is unbounded in every drawing =-=[1, 10, 18]-=-. The best bounds are in our companion paper [10], in which we prove that there exists ∆-regular n-vertex graphs with at least n 1− 8+ɛ ∆+4 slopes in every drawing. Open Problem 5. In all our results,... |

2 | Takao Nishizeki. Rectangular grid drawings of plane graphs - Rahman, Nakano - 1998 |

2 | Takao Nishizeki. A linear algorithm for bend-optimal orthogonal drawings of triconnected cubic plane graphs - Rahman, Nakano - 1999 |

2 | Takao Nishizeki. Box-rectangular drawings of plane graphs - Rahman, Nakano |

2 | Takao Nishizeki. Rectangular drawings of plane graphs without designated corners - Rahman, Nakano |

2 | Takao Nishizeki, and Shubhashis Ghosh. Rectangular drawings of planar graphs - Rahman |

2 |
Ossona de Mendez and Hubert de Fraysseix. Intersection graphs of Jordan arcs
- Patrice
- 1999
(Show Context)
Citation Context ...nar graph is a contact graph of some set of segments with only three distinct slopes. It is an open problem whether every planar graph is the intersection graph of a set of segments in the plane; see =-=[6, 17]-=- for the most recent results. It is even possible that every k-colourable planar graph (k ≤ 4) is the intersection graph of some set of segments using only k distinct slopes. 3s1.1 Definitions We cons... |

1 | Intersection graphs of Jordan arcs
- MENDEZ, FRAYSSEIX
- 1999
(Show Context)
Citation Context ...nar graph is a contact graph of some set of segments with only three distinct slopes. It is an open problem whether every planar graph is the intersection graph of a set of segments in the plane; see =-=[6, 17]-=- for the most recent results. It is even possible that every k-colourable planar graph (k ≤ 4) is the intersection graph of some set of segments using only k distinct slopes. 1.1 DEFINITIONS We consid... |

1 |
Graph drawings with few slopes. Submitted
- Dujmović, Suderman, et al.
- 2005
(Show Context)
Citation Context ...s result independently due to Wagner [26] and Fáry [12]. In this paper we prove lower and upper bounds on the minimum number of segments and slopes in (plane) drawings of graphs. In a companion paper =-=[10]-=-, we consider drawings of non-planar graphs with few slopes. A summary of our results is given in Table 1. A number of comments are in order when considering these results: • The minimum number of slo... |