## Simultaneous embedding of planar graphs with few bends (2004)

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Venue: | In 12th Symposium on Graph Drawing (GD |

Citations: | 26 - 6 self |

### BibTeX

@INPROCEEDINGS{Erten04simultaneousembedding,

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

title = {Simultaneous embedding of planar graphs with few bends},

booktitle = {In 12th Symposium on Graph Drawing (GD},

year = {2004},

pages = {195--205},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossing-free. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossings-free, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixed-edges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for tree-path pairs with at most one bend per tree-edge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).

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Citation Context ... anything about the resolution of the drawing and thus are not well-suited for automated graph drawing. The vertex resolution problem was addressed by de Fraysseix, Pach and Pollack [10] and Schnyder =-=[23]-=- who showed that any n-vertex planar graph can be drawn with straight-lines and no crossings using O(n 2 ) area, with vertices placed at integer grid points. The problem of simultaneos geometric embed... |

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27 |
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Citation Context ...lve thorugh time. In addition to generalizing the notion of planarity, techniques for simultaneous embedding of cycles have been used to show that degree-4 graphs have geometric thickness at most two =-=[12]-=-. Visualization of related graphs, that is, graphs that are defined on the same set of vertices, arise in many different settings. Software engineering, databases, and social network analysis, are all... |

22 | Simultaneous graph drawing: Layout algorithms and visualization schemes
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Citation Context ...illars is shown in [3]. Counter-examples for pairs of general planar graphs, pairs of outerplanar graphs, and triples of paths are also presented there. Modified force-directed algorithms are used in =-=[1, 9]-=- to simultaneously visualize general graphs, while attempting to preserve the user’s mental map and obtaining readable individual drawings. Dynamic graph drawing techniques address some of the problem... |

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16 |
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Citation Context ... or more graphs is more recent. It is known that a simultaneous geometric embedding of an n vertex 3-connected planar graph and its dual can be found in O(n) time using O(n 2 ) area [13]. Brass et al =-=[3]-=- describe linear time algorithms for simultaneous geometric embeddings of pairs of paths, cycles, and caterpillars, also using O(n 2 ) area; see Fig. 1 Cappos and Kobourov [4] show how to simultaneous... |

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Citation Context ...c embedding of two or more graphs is more recent. It is known that a simultaneous geometric embedding of an n vertex 3-connected planar graph and its dual can be found in O(n) time using O(n 2 ) area =-=[13]-=-. Brass et al [3] describe linear time algorithms for simultaneous geometric embeddings of pairs of paths, cycles, and caterpillars, also using O(n 2 ) area; see Fig. 1 Cappos and Kobourov [4] show ho... |

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Citation Context ...s have geometric thickness two. While the thickness and simultaneous embedding problems are related, results from one do not necessarily translate into the other. Bose, Hurtado, Rivera-Campo and Wood =-=[2]-=- show that the complete convex graph K2n can be partitioned into n plane spanning trees and moreover, characterize all the different partitions. In particular, they show that K2n can be partitioned in... |

2 |
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Citation Context ...netic trees are used to visualize the ancestral relationship among groups of species. Depending on the assumptions made, different algorithms produce different phylogenetic trees. Klingner and Amenta =-=[18]-=- and Munzner et al [20] present techniques for visualization of such trees. Comparing the outputs and determining the most likely evolutionary hypothesis can be difficult if the drawings of the trees ... |

2 |
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(Show Context)
Citation Context ...or more graphs is more recent. It is known that a simultaneous geometric embedding of an n vertex 3-connected planar graph and its dual can be found in O(n) time using O(n 2 ) area [13]. Brass et al. =-=[3]-=- describe linear time algorithms for simultaneous geometric embeddings of pairs of paths, cycles, and caterpillars, also using O(n 2 ) area; see Fig. 1. Cappos and Kobourov [4] show how to simultaneou... |

2 |
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(Show Context)
Citation Context ..., do not guarantee anything about the resolution of the drawing and thus are not well-suited for automated graph drawing. The vertex resolution problem was addressed by de Fraysseix, Pach and Pollack =-=[10]-=- and Schnyder [23] who showed that any n-vertex planar graph can be drawn with straight-line segments and no crossings using O(n 2 ) area, with vertices placed at integer grid points. The problem of s... |

1 | Trees on tracks - Cappos, Kobourov - 2004 |

1 |
de Panne. Contour tree simplification with local geometric measures
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- 2004
(Show Context)
Citation Context ... contour trees for scientific and medical visualization. Contour tree simplification applies the ideas of topological persistence to trees and is another application for simultaneous drawing of trees =-=[5]-=-. Simultaneous embedding techniques are also useful in the visualization of graphs that evolve through time, for example, in the context of visualization of the evolution of software [9]. Consider the... |