## Graph Minors and Graphs on Surfaces (2001)

Citations: | 8 - 3 self |

### BibTeX

@MISC{Mohar01graphminors,

author = {Bojan Mohar},

title = {Graph Minors and Graphs on Surfaces},

year = {2001}

}

### OpenURL

### Abstract

Graph minors and the theory of graphs embedded in surfaces are

### Citations

370 | Graph minors. II. Algorithmic aspects of tree-width - Robertson, Seymour - 1986 |

256 |
Graph Minors XIII. The Disjoint Paths Problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ... Robertson and Seymour’s theory gives an O(n 3 ) algorithm for testing embeddability in S using graph minors [37, 52]. Robertson and Seymour recently improved their O(n 3 ) algorithms to O(n 2 log n) =-=[42, 50, 51]-=-. An embeddability testing algorithm can be extended to an algorithm which also constructs an embedding in polynomial time (with estimated complexity O(n 6 ); see Archdeacon [2]). Mohar [25] (and the ... |

243 |
Algorithms For Drawing Graphs: An Annotated Bibliography
- Battista, Eades, et al.
- 1994
(Show Context)
Citation Context ...an explicit bound on the size of the grid. However, it is not difficult to show that the O(n) ×O(n) grid suffices where n is the number of vertices of G0; see Di Battista, Eades, Tamassia, and Tollis =-=[7]-=- for references.12 Bojan Mohar Let G be a Π-embedded graph. If eg(G, Π) ≥ 1, the face-width fw(G, Π) of Π is the smallest integer r such that G has a Π-noncontractible cycle which is the union of r p... |

235 | Efficient planarity testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...us of graphs of bounded tree-width. It turned out that some of the main ingredients in this proof can also be found in the aforementioned work of Seymour [55]. It is well-known that testing planarity =-=[20]-=-, constructing embeddings in the sphere S0 [12], or finding subgraphs that are subdivisions of Kuratowski graphs [62] can be performed by algorithms whose worst case running time is linear. Although t... |

172 | Graph minors X. Obstructions to tree decomposition - Robertson, Seymour - 1991 |

158 |
Linear time algorithms for NP-hard problems restricted to partial k-trees
- Arnborg, Proskurowski
- 1989
(Show Context)
Citation Context ...h minors is Diestel [14, Chapter 12], while excluded minor theorems are treated in Thomas [57]. Graph minors and tree-width are studied in Reed [28], for tree-width and algorithms we refer to [5] and =-=[6]-=-. Embeddings of graphs in surfaces are treated in Mohar and Thomassen [26]; minors and embeddings are also covered in Robertson and Vitray [54]. The proof of the 1 Invited talk at the 18th British Com... |

124 |
Graph minors. I. Excluding a forest
- Robertson, Seymour
- 1983
(Show Context)
Citation Context ... sum of G1 and G2 of order k. 3 The Excluded Minor Theorem Robertson and Seymour proved that in any infinite sequence G1, G2, G3, . . . of graphs there are indices i < j such that Gi is a minor of Gj =-=[30]-=-–[51].Minors and embeddings 5 This seminal result, which establishes the well-quasi-ordering 2 of graphs with respect to the minor relation, is known as the Graph Minor Theorem. In the proof, one may... |

112 |
A linear algorithm for embedding planar graphs using PQ-trees
- Chiba, Nishizeki, et al.
- 1985
(Show Context)
Citation Context ...ut that some of the main ingredients in this proof can also be found in the aforementioned work of Seymour [55]. It is well-known that testing planarity [20], constructing embeddings in the sphere S0 =-=[12]-=-, or finding subgraphs that are subdivisions of Kuratowski graphs [62] can be performed by algorithms whose worst case running time is linear. Although the construction of minimum genus embeddings is ... |

102 | Graph minors. XX. Wagner’s conjecture - Robertson, Seymour |

68 |
Tree width and tangles: A new connectivity measure and some applications
- REED
- 1997
(Show Context)
Citation Context ...xts that cover this subject. A good introduction to graph minors is Diestel [14, Chapter 12], while excluded minor theorems are treated in Thomas [57]. Graph minors and tree-width are studied in Reed =-=[28]-=-, for tree-width and algorithms we refer to [5] and [6]. Embeddings of graphs in surfaces are treated in Mohar and Thomassen [26]; minors and embeddings are also covered in Robertson and Vitray [54]. ... |

59 |
The graph genus problem is NP-complete
- Thomassen
- 1989
(Show Context)
Citation Context ... Kuratowski graphs [62] can be performed by algorithms whose worst case running time is linear. Although the construction of minimum genus embeddings is NP-hard (byMinors and embeddings 11 Thomassen =-=[58]-=-), Filotti, Miller, and Reif [16] proved that for every fixed surface S, there is a polynomial time algorithm for embedding graphs in S. For every fixed surface S, Robertson and Seymour’s theory gives... |

58 |
Tutte, A contribution to the theory of chromatic polynomials
- T
- 1954
(Show Context)
Citation Context ...G) − ε ≤ φc(G ∗ ) ≤ χc(G), where G ∗ is the geometric dual graph of G in S. Proof (sketch). The second inequality can be proved in the same way as the well-known flow-coloring duality result of Tutte =-=[61]-=-, and so we sketch only the proof of the first inequality. Suppose that G is a graph embedded in Sg and that its dual graph G ∗ admits a circular r-flow. If the edge-width of G is w, there is a graph ... |

57 |
Graph minors
- ROBERTSON, SEYMOUR
- 1984
(Show Context)
Citation Context ...combined with two other important results in the Robertson-Seymour theory, that graphs of large tree-width contain large grid minors [34], and that graphs of bounded tree-width are well-quasi-ordered =-=[33]-=-. For the former of these two results, a short proof with constructive bounds was obtained by Diestel, Gorbunov, Jensen, and Thomassen. Theorem 4.4 (Diestel, Gorbunov, Jensen, Thomassen [15]) Let r, m... |

57 |
Graph Minors XVI. Excluding a Non-Planar Graph
- Robertson, Seymour
(Show Context)
Citation Context ...nected. Robertson and Seymour used graph minors to prove a generalization of the Kuratowski Theorem to arbitrary surfaces [37], while they also need surface embeddings in their Excluded Minor Theorem =-=[45]-=-. Various recent results related to graph minors and graphs on surfaces are presented. 1 Introduction A graph H is a minor of another graph G if H can be obtained from a subgraph of G by contracting e... |

56 |
Graph minors | a survey
- Robertson, Seymour
- 1985
(Show Context)
Citation Context ...Robertson and Vitray [54]. The proof of the 1 Invited talk at the 18th British Combinatorial Conference, Sussex, UK, July 2001 12 Bojan Mohar Graph Minor Theorem is sketched in Robertson and Seymour =-=[29]-=-, and a more recent survey with focus on the related disjoint paths problem is [52]. 2 Basic definitions It is convenient to view minors as substructures. Then, a subgraph ¯ H of G is said to be an H-... |

54 | A linear time algorithm for embedding graphs in an arbitrary surface
- Mohar
- 1999
(Show Context)
Citation Context ...rb0(S). A constructive proof for the case of nonorientable surfaces was obtained by Archdeacon and Huneke [4], while the first constructive proof for orientable surfaces appeared just recently (Mohar =-=[25]-=-). An independent constructive proof based on graph minors was also obtained by Seymour [55]. Seymour’s bound on the size of graphs in Forb0(S) is 22(3g+9)9, where g is the Euler genus of S. This numb... |

50 |
A Kuratowski theorem for the projective plane
- Archdeacon
- 1981
(Show Context)
Citation Context ...rtunately, the complete list of graphs in Forb0(S) is known only for the 2-sphere, where Forb0(S0) = {K5, K3,3}, and for the projective plane N1, where there are precisely 35 minimal forbidden minors =-=[18, 1]-=-. The original proof of Theorem 4.1 by Robertson and Seymour is nonconstructive in the sense that it does not provide a bound on the number or the size of graphs in Forb0(S). A constructive proof for ... |

50 |
Excluding a planar graph
- unknown authors
- 1986
(Show Context)
Citation Context ...then prove that these graphs have a special structure. In particular, if G1 is a forest, then the graphs have bounded path-width [30]. If G1 is a planar graph, then the graphs have bounded tree-width =-=[34]-=-. It takes a lot of work to reach the Excluded Minor Theorem 3.1 [45] which describes the structure of the sequence when a more general graph is an excluded minor. To express this result, an additiona... |

47 | Graph minors. IV. Tree-width and well-quasi-ordering - Robertson, Seymour - 1990 |

46 |
Circular chromatic number: a survey, Discrete Mathematics
- Zhu
(Show Context)
Citation Context ...s the smallest real number r such that there exists a real-valued function c : V (G) → [0, r) such that for every edge uv of G, 1 ≤ |c(u) − c(v)| ≤ r − 1. We refer to the recent survey article by Zhu =-=[64]-=- for additional details on circular colorings and flows. Theorem 5.6 (Devos, Goddyn, Mohar, Vertigan, Zhu [13]) There exists a function w : R + ×N → N such that the following holds. If ε > 0 is a real... |

44 | Graph Theory (Second Edition - Diestel - 2000 |

44 |
Graph Minors VII. Disjoint paths on a surface
- Robertson, Seymour
- 1998
(Show Context)
Citation Context ...n of r paths each of which is contained in a single Π-facial walk. If g(G, Π) = 0, we let fw(G, Π) = ∞. Theorem 5.1 has the following analogue for general surfaces. Theorem 5.2 (Robertson and Seymour =-=[36]-=-) Let G0 be a graph that is Π0-embedded in a surface S = S0. Then there is a constant k such that for any graph G which is Π-embedded in S with face-width at least k, (G0, Π0) is a surface minor of (... |

41 |
103 Graphs That Are Irreducible for the Projective Plane
- Glover, Huneke, et al.
- 1979
(Show Context)
Citation Context ...rtunately, the complete list of graphs in Forb0(S) is known only for the 2-sphere, where Forb0(S0) = {K5, K3,3}, and for the projective plane N1, where there are precisely 35 minimal forbidden minors =-=[18, 1]-=-. The original proof of Theorem 4.1 by Robertson and Seymour is nonconstructive in the sense that it does not provide a bound on the number or the size of graphs in Forb0(S). A constructive proof for ... |

39 |
Graph minors. VIII. A Kuratowski theorem for general surfaces
- Robertson, Seymour
- 1990
(Show Context)
Citation Context ...inors and the theory of graphs embedded in surfaces are fundamentally interconnected. Robertson and Seymour used graph minors to prove a generalization of the Kuratowski Theorem to arbitrary surfaces =-=[37]-=-, while they also need surface embeddings in their Excluded Minor Theorem [45]. Various recent results related to graph minors and graphs on surfaces are presented. 1 Introduction A graph H is a minor... |

38 | Branch-width and well-quasi-ordering in matroids and graphs
- Geelen, Gerards, et al.
(Show Context)
Citation Context ...he proof is lengthy and technical as it provides general machinery for the graph minor theory. A8 Bojan Mohar shorter direct proof of this result was recently obtained by Geelen, Gerards and Whittle =-=[17]-=-. In the sequel we give a new, much simpler proof of this result restricted to graphs in Forb0(S). Theorem 4.5 Let g and w be positive integers and let S be a surface of Euler genus g. Then there is a... |

32 |
Depth-first search and Kuratowski subgraphs
- Williamson
- 1984
(Show Context)
Citation Context ...n the aforementioned work of Seymour [55]. It is well-known that testing planarity [20], constructing embeddings in the sphere S0 [12], or finding subgraphs that are subdivisions of Kuratowski graphs =-=[62]-=- can be performed by algorithms whose worst case running time is linear. Although the construction of minimum genus embeddings is NP-hard (byMinors and embeddings 11 Thomassen [58]), Filotti, Miller,... |

29 | C.: Highly connected sets and the excluded grid theorem
- Diestel, Jensen, et al.
- 1999
(Show Context)
Citation Context ...i-ordered [33]. For the former of these two results, a short proof with constructive bounds was obtained by Diestel, Gorbunov, Jensen, and Thomassen. Theorem 4.4 (Diestel, Gorbunov, Jensen, Thomassen =-=[15]-=-) Let r, m be positive integers, and let G be a graph of tree-width at least r 4m2 (r+2) . Then G contains either Km or the r × r grid as a minor. The second result, the well-quasi-ordering of graphs ... |

29 | Graph minors. IX. Disjoint crossed paths - Robertson, Seymour - 1990 |

25 | Ecient planarity testing - Hopcroft, Tarjan - 1974 |

21 |
A Kuratowski theorem for non-orientable surfaces
- Archdeacon, Huneke
- 1989
(Show Context)
Citation Context ...uctive in the sense that it does not provide a bound on the number or the size of graphs in Forb0(S). A constructive proof for the case of nonorientable surfaces was obtained by Archdeacon and Huneke =-=[4]-=-, while the first constructive proof for orientable surfaces appeared just recently (Mohar [25]). An independent constructive proof based on graph minors was also obtained by Seymour [55]. Seymour’s b... |

21 |
A simpler proof of the excluded minor theorem for higher surfaces
- Thomassen
- 1997
(Show Context)
Citation Context ... has less than 30 vertices. In the late 90’s, Thomassen observed the possibility of obtaining a short proof of Theorem 4.1. He found a very short proof of the following result. Theorem 4.3 (Thomassen =-=[60]-=-) Let G ∈ Forb(Sg). Then G contains no k × k grid as a minor, where k = ⌈3300g 3/2 ⌉. Theorem 4.3 implies Theorem 4.1 when combined with two other important results in the Robertson-Seymour theory, th... |

19 |
Typical Subgraphs of 3- and 4-Connected Graphs
- Oporowski, Oxley, et al.
- 1993
(Show Context)
Citation Context ...re 2 shows a graph G, a tree decomposition of width 3, and the underlying tree T. Let us observe that the graph G is outerplanar and hence it also has a tree decomposition of width 2. It was shown in =-=[27]-=- that if a graph G has a tree decomposition of width at most w, then G has a tree decomposition of width at most w that further satisfies: (T3) For every two vertices t, t ′ of T and every positive in... |

17 |
Representativity of surface embeddings, In: Paths, Flows and VLSI-Layout
- Robertson, Vitray
- 1990
(Show Context)
Citation Context ...d [28], for tree-width and algorithms we refer to [5] and [6]. Embeddings of graphs in surfaces are treated in Mohar and Thomassen [26]; minors and embeddings are also covered in Robertson and Vitray =-=[54]-=-. The proof of the 1 Invited talk at the 18th British Combinatorial Conference, Sussex, UK, July 2001 12 Bojan Mohar Graph Minor Theorem is sketched in Robertson and Seymour [29], and a more recent s... |

14 | Graph minors. XVII. Taming a vortex - Robertson, Seymour - 1999 |

14 |
Embeddings and minors
- Thomassen
- 1995
(Show Context)
Citation Context ... information on some of the most appealing results about graph minors and their relation to the study of graphs on surfaces. Besides a stimulating survey article on minors and embeddings by Thomassen =-=[59]-=-, there are numerous existing texts that cover this subject. A good introduction to graph minors is Diestel [14, Chapter 12], while excluded minor theorems are treated in Thomas [57]. Graph minors and... |

13 |
Projective planarity in linear time
- Mohar
- 1993
(Show Context)
Citation Context ... embedding of G in S, or (b) finds a subgraph K ⊆ G which is a subdivision of some graph in Forb(S). A simpler linear time algorithm for embedding graphs in the projective plane is described by Mohar =-=[23]-=-, while a simpler algorithm for the torus was developed recently by Juvan and Mohar [21]. 5 Surface minors and the face-width Given a Π-embedded graph G, every minor H of G can be considered as being ... |

11 | Graph Minors XXII: irrelevant vertices in linkage problems”, preprint
- Robertson, Seymour
- 1992
(Show Context)
Citation Context ...of G1 and G2 of order k. 3 The Excluded Minor Theorem Robertson and Seymour proved that in any infinite sequence G1, G2, G3, . . . of graphs there are indices i < j such that Gi is a minor of Gj [30]–=-=[51]-=-.Minors and embeddings 5 This seminal result, which establishes the well-quasi-ordering 2 of graphs with respect to the minor relation, is known as the Graph Minor Theorem. In the proof, one may assu... |

10 |
Densely embedded graphs
- Archdeacon
- 1992
(Show Context)
Citation Context ... of theses results are treated in Seymour and Thomas [56] and Mohar [24] who proved that face-width of order O(g log g) (g = eg(G, Π)) is sufficient, and this is essentially best possible (Archdeacon =-=[3]-=-). There are numerous other results where Theorem 5.2 is used. However, the most surprising seems to be the flow-coloring duality on general surfaces discovered recently by Devos, Goddyn, Mohar, Verti... |

10 |
Separating and nonseparating disjoint homotopic cycles in graph embeddings
- Brunet, Mohar, et al.
- 1996
(Show Context)
Citation Context ...ws from Theorem 5.2 that large face-width forces the existence of noncontractible surface separating cycles (where “large” may depend on the surface). Zha and Zhao [63] and Brunet, Mohar, and Richter =-=[11]-=- proved that face-width 6 (even 5 for nonorientable surfaces) is sufficient. If Conjecture 5.3 is true, also the following may hold as suggested in Mohar and Thomassen [26]. Conjecture 5.5 Let T be a ... |

10 |
On determining the genus of a graph
- Filotti, Miller, et al.
- 1979
(Show Context)
Citation Context ...rformed by algorithms whose worst case running time is linear. Although the construction of minimum genus embeddings is NP-hard (byMinors and embeddings 11 Thomassen [58]), Filotti, Miller, and Reif =-=[16]-=- proved that for every fixed surface S, there is a polynomial time algorithm for embedding graphs in S. For every fixed surface S, Robertson and Seymour’s theory gives an O(n 3 ) algorithm for testing... |

10 | Graph Minors XII. Distance on a Surface - Robertson, Seymour - 1995 |

10 |
A survey of linkless embeddings, in: Graph Structure Theory
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ...ated to embeddings in various topological spaces. Such examples include graphs embeddable in a fixed surface, graphs embeddable in R 3 in some specific way, for instance, linklessly embeddable graphs =-=[53, 57]-=- (i.e., graphs which admit an embedding in R 3 such that no two disjoint cycles of the graph are linked in R 3 ), knotlessly embeddable graphs (every cycle of the graph is embedded as an unknot), etc.... |

9 | Graph Minors XXI. Graphs with unique linkages
- Robertson, Seymour
(Show Context)
Citation Context ... Robertson and Seymour’s theory gives an O(n 3 ) algorithm for testing embeddability in S using graph minors [37, 52]. Robertson and Seymour recently improved their O(n 3 ) algorithms to O(n 2 log n) =-=[42, 50, 51]-=-. An embeddability testing algorithm can be extended to an algorithm which also constructs an embedding in polynomial time (with estimated complexity O(n 6 ); see Archdeacon [2]). Mohar [25] (and the ... |

9 | Recent excluded minor theorems for graphs
- Thomas
- 1999
(Show Context)
Citation Context ...ings by Thomassen [59], there are numerous existing texts that cover this subject. A good introduction to graph minors is Diestel [14, Chapter 12], while excluded minor theorems are treated in Thomas =-=[57]-=-. Graph minors and tree-width are studied in Reed [28], for tree-width and algorithms we refer to [5] and [6]. Embeddings of graphs in surfaces are treated in Mohar and Thomassen [26]; minors and embe... |

8 |
Grid minors of graphs on the torus
- Graaf, Schrijver
- 1994
(Show Context)
Citation Context ...flow r+ε r ϕ0. As the face-width of Gi is large enough, Theorem 5.2 can be used to show that the toroidal q × q grid Rq is a surface minor in Gi, where q = ⌈2r2 /ε⌉. (As proved by Graaf and Schrijver =-=[19]-=-, it is sufficient that the face-width is ≥ 3q + 3.) The toroidal grid consists of pairwise disjoint “vertical” cycles 2 A1, . . .,Aq and pairwise disjoint “horizontal” cycles B1, . . .,Bq. Let Dkl be... |

8 |
Uniqueness and minimality of large face-width embeddings of graphs, Combinatorica 15
- Mohar
- 1995
(Show Context)
Citation Context ...ing of G in S is unique. Consequently, sufficiently large facewidth of a 3-connected graph implies uniqueness of the embedding. Both of theses results are treated in Seymour and Thomas [56] and Mohar =-=[24]-=- who proved that face-width of order O(g log g) (g = eg(G, Π)) is sufficient, and this is essentially best possible (Archdeacon [3]). There are numerous other results where Theorem 5.2 is used. Howeve... |

6 | Graph minors XVIII. Tree-decompositions and well-quasiordering - Robertson, Seymour |

5 | Graph minors. XIV. Extending an embedding - Robertson, Seymour - 1995 |

5 | Graph minors. XV. Giant steps - Robertson, Seymour - 1996 |

5 | Graph minors. XIX. Well-quasi-ordering on a surface, preprint - Robertson, Seymour - 1989 |