## A Simpler and Faster Torus Embedding Algorithm (2006)

### BibTeX

@MISC{Woodcock06asimpler,

author = {Jennifer Roselynn Woodcock},

title = {A Simpler and Faster Torus Embedding Algorithm},

year = {2006}

}

### OpenURL

### Abstract

### Citations

1264 |
Graph Theory with Applications
- Bondy, Murty
- 1976
(Show Context)
Citation Context ... to find all of the torus obstructions. 2sChapter 2 Background Elementary concepts and definitions in graph theory and their proofs can be found in introductory texts by West [35] and Bondy and Murty =-=[7]-=-. Archdeacon’s survey of topological graph theory gives an excellent introduction to the study of graph embeddings [3]. More in-depth discussions can be found in Henle’s book A Combinatorial Introduct... |

820 |
Introduction to Graph Theory
- West
- 1996
(Show Context)
Citation Context ...ur algorithm and using it to find all of the torus obstructions. 2sChapter 2 Background Elementary concepts and definitions in graph theory and their proofs can be found in introductory texts by West =-=[35]-=- and Bondy and Murty [7]. Archdeacon’s survey of topological graph theory gives an excellent introduction to the study of graph embeddings [3]. More in-depth discussions can be found in Henle’s book A... |

511 |
The complexity of computing the permanent
- Valiant
- 1979
(Show Context)
Citation Context ...g a fast and correct torus embedding algorithm, in turn, has algorithmic implications as there are computationally intractable problems that can be solved in polynomial time for toroidal graphs (e.g. =-=[16, 33]-=-). 16sChapter 3 The Generic and Planar Embedding Algorithms In this chapter, we first present a generic backtracking approach for embedding graphs on orientable surfaces. Then, we describe the modific... |

468 |
Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...story and Motivation Many complex linear time algorithms for embedding graphs on the plane exist, including those of Hopcroft and Tarjan (this was the first, developed in 1974) [19], Booth and Lueker =-=[8]-=-, Fraysseix and Rosentiehl [12], Williamson [36, 37], and Boyer and Myrvold [9]. Less complex are the O(n 2 ) algorithms of Klotz [23] and of Demoucron, Malgrange, and Pertuiset [13]. The latter of th... |

226 | Tarjan, Efficient planarity testing
- Hopcroft, E
- 1974
(Show Context)
Citation Context ...is not toroidal. 2.3 History and Motivation Many complex linear time algorithms for embedding graphs on the plane exist, including those of Hopcroft and Tarjan (this was the first, developed in 1974) =-=[19]-=-, Booth and Lueker [8], Fraysseix and Rosentiehl [12], Williamson [36, 37], and Boyer and Myrvold [9]. Less complex are the O(n 2 ) algorithms of Klotz [23] and of Demoucron, Malgrange, and Pertuiset ... |

200 |
Graphs on surfaces
- Mohar, Thomassen
- 2001
(Show Context)
Citation Context ...ction to the study of graph embeddings [3]. More in-depth discussions can be found in Henle’s book A Combinatorial Introduction to Topology [18] and in Mohar’s and Thomassen’s book Graphs on Surfaces =-=[26]-=-. Here we present basic definitions and key concepts pertinent to understanding the material presented in this thesis followed by a history of related and relevant work that provided motivation for th... |

165 |
Sur le problème des courbes gauches en topologie, Fund
- Kuratowski
(Show Context)
Citation Context ...faces, and, more specifically, the problem of finding the complete set of torus obstructions. Kuratowski proved that there are there are two (minor order [34]) obstructions for the plane, K5 and K3,3 =-=[24]-=-, and gave Theorem 2.3.1, now known as Kuratowski’s Theorem. Theorem 2.3.1. [24, 34] Kuratowski’s Theorem A graph is planar if and only if it does not contain a subgraph homeomorphic to K5 or K3,3. Th... |

81 |
A Combinatorial Introduction to Topology
- Henle
- 1979
(Show Context)
Citation Context ...urvey of topological graph theory gives an excellent introduction to the study of graph embeddings [3]. More in-depth discussions can be found in Henle’s book A Combinatorial Introduction to Topology =-=[18]-=- and in Mohar’s and Thomassen’s book Graphs on Surfaces [26]. Here we present basic definitions and key concepts pertinent to understanding the material presented in this thesis followed by a history ... |

49 |
A Kuratowski theorem for the projective plane
- Archdeacon
- 1981
(Show Context)
Citation Context ... is the projective plane; Glover, Huneke, and Wang listed 103 topological obstructions and 35 minor-order obstructions for the projective plane [17] and Archdeacon proved that this is a complete list =-=[1]-=-. Archdeacon also found the complete set of 21 3-regular topological obstructions for the spindle, the surface formed by identifying two points on the sphere [4]. Finding the complete obstruction set ... |

40 |
103 graphs that are irreducible for the projective plane
- Glover, Huneke, et al.
- 1979
(Show Context)
Citation Context ...he plane for which the complete obstruction set is known is the projective plane; Glover, Huneke, and Wang listed 103 topological obstructions and 35 minor-order obstructions for the projective plane =-=[17]-=- and Archdeacon proved that this is a complete list [1]. Archdeacon also found the complete set of 21 3-regular topological obstructions for the spindle, the surface formed by identifying two points o... |

39 |
P.: Graph minors. VIII. A Kuratowski Theorem for General Surfaces
- Robertson, Seymour
- 1990
(Show Context)
Citation Context ... Bodendiek and Wagner for orientable surfaces [6], by Archdeacon and Huneke for non-orientable surfaces [2], and independently by Robertson and Seymour for both orientable and non-orientable surfaces =-=[31]-=-. This fact leads to Theorem 2.3.2, a generalization of Kuratowski’s Theorem to surfaces other than the plane. Theorem 2.3.2. [6, 2, 31] Generalization of Kuratowski’s Theorem A graph is embeddable on... |

32 |
Depth-first search and Kuratowski subgraphs
- Williamson
- 1984
(Show Context)
Citation Context ...lgorithms for embedding graphs on the plane exist, including those of Hopcroft and Tarjan (this was the first, developed in 1974) [19], Booth and Lueker [8], Fraysseix and Rosentiehl [12], Williamson =-=[36, 37]-=-, and Boyer and Myrvold [9]. Less complex are the O(n 2 ) algorithms of Klotz [23] and of Demoucron, Malgrange, and Pertuiset [13]. The latter of these provided the inspiration for the torus embedding... |

30 | On the theory of Pfaffian orientations I. Perfect matchings and permanents
- Galluccio, Loebl
(Show Context)
Citation Context ...g a fast and correct torus embedding algorithm, in turn, has algorithmic implications as there are computationally intractable problems that can be solved in polynomial time for toroidal graphs (e.g. =-=[16, 33]-=-). 16sChapter 3 The Generic and Planar Embedding Algorithms In this chapter, we first present a generic backtracking approach for embedding graphs on orientable surfaces. Then, we describe the modific... |

24 | Stop minding your P’s and Q’s: A simplified O(n) planar embedding algorithm
- Boyer, Myrvold
- 1999
(Show Context)
Citation Context ...n the plane exist, including those of Hopcroft and Tarjan (this was the first, developed in 1974) [19], Booth and Lueker [8], Fraysseix and Rosentiehl [12], Williamson [36, 37], and Boyer and Myrvold =-=[9]-=-. Less complex are the O(n 2 ) algorithms of Klotz [23] and of Demoucron, Malgrange, and Pertuiset [13]. The latter of these provided the inspiration for the torus embedding algorithm presented in thi... |

21 |
A Kuratowski theorem for nonorientable surfaces
- Archdeacon, Huneke
- 1989
(Show Context)
Citation Context ... or K3,3. That the number of obstructions is finite for any surface of fixed genus was proved by Bodendiek and Wagner for orientable surfaces [6], by Archdeacon and Huneke for non-orientable surfaces =-=[2]-=-, and independently by Robertson and Seymour for both orientable and non-orientable surfaces [31]. This fact leads to Theorem 2.3.2, a generalization of Kuratowski’s Theorem to surfaces other than the... |

20 |
An approach to the subgraph homeomorphism problem
- Asano
- 1985
(Show Context)
Citation Context ...onstant number of planarity tests to determine if G is toroidal if G has no subgraph homeomorphic to K3,3 [15]. The “transformation” of K5 to K3,3 can be done in linear time using the method of Asano =-=[5]-=-. As such, incorporating this transformation into our algorithm would not improve upon its exponential running time. However, we expect that it would 37sdecrease the running time in practice in most c... |

20 |
A depth-first search characterization of planarity
- Fraysseix, Rosenstiehl
- 1982
(Show Context)
Citation Context ...lex linear time algorithms for embedding graphs on the plane exist, including those of Hopcroft and Tarjan (this was the first, developed in 1974) [19], Booth and Lueker [8], Fraysseix and Rosentiehl =-=[12]-=-, Williamson [36, 37], and Boyer and Myrvold [9]. Less complex are the O(n 2 ) algorithms of Klotz [23] and of Demoucron, Malgrange, and Pertuiset [13]. The latter of these provided the inspiration fo... |

18 |
Embedding graphs in the plane – algorithmic aspects
- Williamson
(Show Context)
Citation Context ...lgorithms for embedding graphs on the plane exist, including those of Hopcroft and Tarjan (this was the first, developed in 1974) [19], Booth and Lueker [8], Fraysseix and Rosentiehl [12], Williamson =-=[36, 37]-=-, and Boyer and Myrvold [9]. Less complex are the O(n 2 ) algorithms of Klotz [23] and of Demoucron, Malgrange, and Pertuiset [13]. The latter of these provided the inspiration for the torus embedding... |

13 |
Projective planarity in linear time
- Mohar
- 1993
(Show Context)
Citation Context ... of these provided the inspiration for the torus embedding algorithm presented in this thesis. For embedding graphs on the projective plane, there is a complex linear time algorithm designed by Mohar =-=[25]-=-, and a less complex O(n 2 ) algorithm designed and implemented by Myrvold and 13sAlgorithm 2.2 Torus Obstruction(graph G) 1: if G has minimum degree less than three then 2: Halt: G has minimum degree... |

10 | Practical toroidality testing
- Myrvold, Neufeld
- 1997
(Show Context)
Citation Context ...et been successfully implemented and it is possible that their complexity will be prohibitive to their practicality [27]. An exponential torus embedding algorithm was developed by Myrvold and Neufeld =-=[30, 29]-=- and enhanced by Chambers [10], and is practical for small graphs. Filotti also presented a specialized algorithm for embedding only 3-regular graphs on the torus [14] which he claimed to have polynom... |

9 |
Solution to König’s graph embedding problem
- Bodendiek, Wagner
- 1989
(Show Context)
Citation Context ... only if it does not contain a subgraph homeomorphic to K5 or K3,3. That the number of obstructions is finite for any surface of fixed genus was proved by Bodendiek and Wagner for orientable surfaces =-=[6]-=-, by Archdeacon and Huneke for non-orientable surfaces [2], and independently by Robertson and Seymour for both orientable and non-orientable surfaces [31]. This fact leads to Theorem 2.3.2, a general... |

8 | Embedding graphs containing K5-subdivisions
- Gagarin, Kocay
- 2002
(Show Context)
Citation Context ... to either: • find a subgraph homeomorphic to K3,3 in G, if one exists, or • perform a small constant number of planarity tests to determine if G is toroidal if G has no subgraph homeomorphic to K3,3 =-=[15]-=-. The “transformation” of K5 to K3,3 can be done in linear time using the method of Asano [5]. As such, incorporating this transformation into our algorithm would not improve upon its exponential runn... |

7 |
Über einer eigenschaft der ebener complexe
- Wagner
- 1937
(Show Context)
Citation Context ...nding the complete set of obstructions for surfaces, and, more specifically, the problem of finding the complete set of torus obstructions. Kuratowski proved that there are there are two (minor order =-=[34]-=-) obstructions for the plane, K5 and K3,3 [24], and gave Theorem 2.3.1, now known as Kuratowski’s Theorem. Theorem 2.3.1. [24, 34] Kuratowski’s Theorem A graph is planar if and only if it does not con... |

4 |
Topological graph theory: a survey. Congressus Numerantium
- Archdeacon
- 1996
(Show Context)
Citation Context ... their proofs can be found in introductory texts by West [35] and Bondy and Murty [7]. Archdeacon’s survey of topological graph theory gives an excellent introduction to the study of graph embeddings =-=[3]-=-. More in-depth discussions can be found in Henle’s book A Combinatorial Introduction to Topology [18] and in Mohar’s and Thomassen’s book Graphs on Surfaces [26]. Here we present basic definitions an... |

4 |
Hunting for torus obstructions
- Chambers
- 2002
(Show Context)
Citation Context ...nd it is possible that their complexity will be prohibitive to their practicality [27]. An exponential torus embedding algorithm was developed by Myrvold and Neufeld [30, 29] and enhanced by Chambers =-=[10]-=-, and is practical for small graphs. Filotti also presented a specialized algorithm for embedding only 3-regular graphs on the torus [14] which he claimed to have polynomial running time, but Myrvold ... |

4 |
An algorithm for imbedding cubic graphs in the torus
- Filotti
- 1980
(Show Context)
Citation Context ...loped by Myrvold and Neufeld [30, 29] and enhanced by Chambers [10], and is practical for small graphs. Filotti also presented a specialized algorithm for embedding only 3-regular graphs on the torus =-=[14]-=- which he claimed to have polynomial running time, but Myrvold and Kocay proved that it is incorrect [28]. Myrvold and Kocay also discuss critical design issues in finding a polynomial time algorithm ... |

4 | Embedding graphs in the torus in linear time
- Juvan, Marincek, et al.
- 1995
(Show Context)
Citation Context ... order obstruction. Roth [32] . For the torus, there currently is no known implementation of an efficient embedding algorithm. Mohar proposed a linear time algorithm for embedding graphs on the torus =-=[21]-=- and Juvan and Mohar simplified the linear time algorithm to create an O(n 3 ) variant [22]. Neither of these algorithms has yet been successfully implemented and it is possible that their complexity ... |

4 |
A simplified algorithm for embedding a graph into the torus, http://www.fmf.uni-lj.si/∼mohar/Algorithms.html
- Juvan
(Show Context)
Citation Context ...f an efficient embedding algorithm. Mohar proposed a linear time algorithm for embedding graphs on the torus [21] and Juvan and Mohar simplified the linear time algorithm to create an O(n 3 ) variant =-=[22]-=-. Neither of these algorithms has yet been successfully implemented and it is possible that their complexity will be prohibitive to their practicality [27]. An exponential torus embedding algorithm wa... |

3 | Bonnington, Obstructions for embedding cubic graphs on the spindle surface
- Archdeacon, P
- 2004
(Show Context)
Citation Context ...on proved that this is a complete list [1]. Archdeacon also found the complete set of 21 3-regular topological obstructions for the spindle, the surface formed by identifying two points on the sphere =-=[4]-=-. Finding the complete obstruction set for the torus is a natural next step in this field of research. The exponential torus embedding algorithm of Myrvold and Neufeld is practical enough to have foun... |

3 |
A constructive proof of Kuratowski’s theorem
- Klotz
- 1989
(Show Context)
Citation Context ...jan (this was the first, developed in 1974) [19], Booth and Lueker [8], Fraysseix and Rosentiehl [12], Williamson [36, 37], and Boyer and Myrvold [9]. Less complex are the O(n 2 ) algorithms of Klotz =-=[23]-=- and of Demoucron, Malgrange, and Pertuiset [13]. The latter of these provided the inspiration for the torus embedding algorithm presented in this thesis. For embedding graphs on the projective plane,... |

3 | Simpler projective plane embedding
- Roth, Myrvold
- 2005
(Show Context)
Citation Context ... where c = Πb(a), marking each record as visited along the way. Algorithm 2.1 gives pseudocode for the face walking algorithm and Myrvold and Roth provide a more in depth discussion of this algorithm =-=[32]-=- including how to modify it for non-orientable surfaces. 2.2.4 Obstructions A topological obstruction for a surface S is a graph G with minimum degree three that is not embeddable on S but for all edg... |

2 |
The obstructions for toroidal graphs with no K3,3’s. 2004 preprint
- Chambers, Gagarin, et al.
(Show Context)
Citation Context ...d the complete set of 270 projective planar torus obstructions [20] and Chambers, Gagarin and Myrvold found the complete set of torus obstructions which do not contain a subgraph homeomorphic to K3,3 =-=[11]-=-. Obviously the sets of obstructions found by researchers overlap in some cases, but each provided significant contributions to the knowledge base about this problem. In all, 239,451 topological obstr... |

2 |
Graphes planaires. Revue Francaise Recherche Operationnelle
- Demoucron, Malgrange, et al.
- 1964
(Show Context)
Citation Context ...n Chapter 3. The new torus embedding algorithm that is the main subject of this thesis was inspired by the quadratic algorithm of Demoucron, Malgrange, and Pertuiset for embedding graphs on the plane =-=[13]-=-. Also in chapter 3, therefore, as a preface to the presentation of our algorithm, we explain how their algorithm fits into the generic framework. In Chapter 4 we discuss the details of our new torus ... |

2 |
Algorithms and obstructions for embedding graphs in the torus
- Juvan
- 1995
(Show Context)
Citation Context ...also used a “Split-Delete” approach on known small obstructions to generate a collection of larger obstructions [10]. Further, Juvan found the complete set of 270 projective planar torus obstructions =-=[20]-=- and Chambers, Gagarin and Myrvold found the complete set of torus obstructions which do not contain a subgraph homeomorphic to K3,3 [11]. Obviously the sets of obstructions found by researchers overl... |

1 |
On Filotti’s algorithm for embedding 3-regular graphs on the torus
- Myrvold, Kocay, et al.
- 2004
(Show Context)
Citation Context ...ilotti also presented a specialized algorithm for embedding only 3-regular graphs on the torus [14] which he claimed to have polynomial running time, but Myrvold and Kocay proved that it is incorrect =-=[28]-=-. Myrvold and Kocay also discuss critical design issues in finding a polynomial time algorithm for embedding graphs on the torus [28]. 14sIn addition to the graph embedding problem in general, the res... |