## An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges (2008)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Fowler08anspqr-tree,

author = {J. Joseph Fowler and Carsten Gutwenger and Michael Jünger and Petra Mutzel and Michael Schulz},

title = {An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges},

year = {2008}

}

### OpenURL

### Abstract

We present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs.

### Citations

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(Show Context)
Citation Context ...mbinatorial embedding of G. Given two planar graphs G1 = (V, E1) and G2 = (V, E2) on the same vertex set V , a simultaneous embedding with fixed edges (SEFE) consists of planar drawings Γi of Gi, i ∈ =-=[1, 2]-=-, such that each vertex is mapped to the same point in the plane for Γ1 and Γ2 and each edge in G1 ∩ G2 is represented by the same simple curve in the plane for both drawings. 3 A Planar Graph, a Cycl... |

93 | An optimal algorithm for Euclidean shortest paths in the plane
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(Show Context)
Citation Context ...y number of edges with x or y as endpoints) are the only graphs to always have a SEFE with any planar graph. Their drawing algorithms are based upon using an optimal Euclidean shortest path algorithm =-=[13]-=-. We also apply this technique in our algorithms. In this paper we examine the pairs of a planar graph G1 with a pseudoforest G2. A SEFE is not always guaranteed unless all non-cycle edges of G2 are i... |

81 | Incremental Planarity Testing
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- 1989
(Show Context)
Citation Context ...a maximal 2-connected subgraph of a graph G. If G is 2-connected, the SPQR-tree T of G represents its decomposition into 3-connected componentscomprising serial, parallel, and 3-connected structures =-=[3]-=-. The respective structure is given by a skeleton graph associated with each tree node which is either a cycle (S-node), a bundle of parallel edges (P-node), or a 3-connected simple graph (R-node); Q-... |

25 | Simultaneous embedding of planar graphs with few bends
- Erten, Kobourov
(Show Context)
Citation Context ...1] while deciding SGE is NP-hard for two graphs [8]. The complexity of deciding SEFE for two graphs is still open. Many approaches have been made to decide the problem for some classes of graph pairs =-=[4,7, 9,10]-=-. Frati [10] showed that trees and planar graphs always have a SEFE. Fowler et al. [9] improved this result to show that forests, circular caterpillars (removal of all degree-1 vertices yields a cycle... |

16 | Simultaneous geometric graph embeddings
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- 2008
(Show Context)
Citation Context ...ultaneous embedding have become an important branch within the field of graph drawing. It is known that deciding SEFE is NP-complete for three graphs [11] while deciding SGE is NP-hard for two graphs =-=[8]-=-. The complexity of deciding SEFE for two graphs is still open. Many approaches have been made to decide the problem for some classes of graph pairs [4,7, 9,10]. Frati [10] showed that trees and plana... |

13 | Simultaneous graph embeddings with fixed edges
- Gassner, Jünger, et al.
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(Show Context)
Citation Context ...ne drawings as well as the other variations of simultaneous embedding have become an important branch within the field of graph drawing. It is known that deciding SEFE is NP-complete for three graphs =-=[11]-=- while deciding SGE is NP-hard for two graphs [8]. The complexity of deciding SEFE for two graphs is still open. Many approaches have been made to decide the problem for some classes of graph pairs [4... |

12 |
Embedding graphs simultaneously with fixed edges
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(Show Context)
Citation Context ...1] while deciding SGE is NP-hard for two graphs [8]. The complexity of deciding SEFE for two graphs is still open. Many approaches have been made to decide the problem for some classes of graph pairs =-=[4,7, 9,10]-=-. Frati [10] showed that trees and planar graphs always have a SEFE. Fowler et al. [9] improved this result to show that forests, circular caterpillars (removal of all degree-1 vertices yields a cycle... |

11 |
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Citation Context ...ing Γ of a planar graph G, a cycle C in G forms a Jordan curve that splits the plane into two connected components. One is bounded by C and the other is unbounded as given by the Jordan curve theorem =-=[14]-=-. We say that some vertex v ∈ G \C lies in the interior (exterior) of C if it is mapped to a position in the bounded (unbounded) component. A combinatorial embedding of a planar graph G is defined as ... |

10 | M.: Orthogonal graph drawing with constraints
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(Show Context)
Citation Context ...diagrams. To do this, common vertices and edges are placed and drawn equally in each drawing. This can be modeled via embedding constraints. Various embedding constraints have already been studied in =-=[2,5, 6]-=-; Gutwenger et al. [12] apply SPQR-trees to efficiently decide if a graph has a combinatorial embedding with respect to a set of hierarchical constraints modeling grouping † Partially supported by the... |

8 | A note on simultaneous embedding of planar graphs
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(Show Context)
Citation Context ...1] while deciding SGE is NP-hard for two graphs [8]. The complexity of deciding SEFE for two graphs is still open. Many approaches have been made to decide the problem for some classes of graph pairs =-=[4,7, 9,10]-=-. Frati [10] showed that trees and planar graphs always have a SEFE. Fowler et al. [9] improved this result to show that forests, circular caterpillars (removal of all degree-1 vertices yields a cycle... |

6 | Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges
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(Show Context)
Citation Context |

6 | Planarity testing and optimal edge insertion with embedding constraints
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(Show Context)
Citation Context ...n vertices and edges are placed and drawn equally in each drawing. This can be modeled via embedding constraints. Various embedding constraints have already been studied in [2,5, 6]; Gutwenger et al. =-=[12]-=- apply SPQR-trees to efficiently decide if a graph has a combinatorial embedding with respect to a set of hierarchical constraints modeling grouping † Partially supported by the German Science Foundat... |

5 | Planar graphs with topological constraints
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(Show Context)
Citation Context ...diagrams. To do this, common vertices and edges are placed and drawn equally in each drawing. This can be modeled via embedding constraints. Various embedding constraints have already been studied in =-=[2,5, 6]-=-; Gutwenger et al. [12] apply SPQR-trees to efficiently decide if a graph has a combinatorial embedding with respect to a set of hierarchical constraints modeling grouping † Partially supported by the... |

2 | Drawing database schemas
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(Show Context)
Citation Context ...diagrams. To do this, common vertices and edges are placed and drawn equally in each drawing. This can be modeled via embedding constraints. Various embedding constraints have already been studied in =-=[2,5, 6]-=-; Gutwenger et al. [12] apply SPQR-trees to efficiently decide if a graph has a combinatorial embedding with respect to a set of hierarchical constraints modeling grouping † Partially supported by the... |