## Recent Excluded Minor Theorems for Graphs (1999)

Venue: | IN SURVEYS IN COMBINATORICS, 1999 267 201-222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8 |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Thomas99recentexcluded,

author = {Robin Thomas},

title = {Recent Excluded Minor Theorems for Graphs},

booktitle = {IN SURVEYS IN COMBINATORICS, 1999 267 201-222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8},

year = {1999},

pages = {201--222},

publisher = {Univ. Press}

}

### OpenURL

### Abstract

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger’s conjecture on t-colorability of graphs with no Kt+1 minor, Tutte’s edge 3-coloring conjecture on edge 3-colorability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colorability of hypergraphs, and sign-nonsingular matrices.

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Citation Context ...ill open. We refer to [62] for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I discuss a structural result pertaining to matching theory. An orientation D of a graph G is Pfaffian =-=[23, 24, 30]-=- if every even circuit C of G such that G\V (C) has a perfect matching has an odd number of edges directed in D in the direction of each orientation of C. The significance of Pfaffian orientations is ... |

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58 |
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(Show Context)
Citation Context ...rallel edges. Agraphisaminor of another if the first can be obtained from a subgraph of the second by contracting edges. An H minor is a minor isomorphic to H. The following is Wagner’s reformulation =-=[75]-=- of Kuratowski’s theorem [27]. Theorem 1.1 A graph is planar if and only if it has no minor isomorphic to K5 or K3,3. Kuratowski’s theorem is important, because it gives a good characterization (in th... |

57 | pfaffian orientations, and even directed circuits
- Robertson, Seymour, et al.
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(Show Context)
Citation Context ...est if a given bipartite graph has a Pfaffian orientation. The next theorem, proven independently by McCuaig [35, 36]sRecent Excluded Minor Theorems for Graphs 16 and by Robertson, Seymour and Thomas =-=[54]-=-, can be used to design such an algorithm. We say that a bipartite graph is a brace if every matching of size at most two can be extended to a perfect matching. An argument similar to the one in the p... |

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(Show Context)
Citation Context ...of of Theorem 10.2 to prove Conjecture 10.4, but no work in that direction has yet been done. Tutte made two other conjectures about nowhere-zero flows, known as the 3-flow [72] and 5flow conjectures =-=[69]-=-. Both of them are still open. We refer to [62] for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I discuss a structural result pertaining to matching theory. An orientation D of a... |

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(Show Context)
Citation Context ...c to K5 if and only if it can be obtained from planar graphs and V8 by means of 0-, 1-, 2- and3-sums. There are many similar results in Graph Theory, known as excluded minor theorems (see for example =-=[5, 6, 16, 19, 27, 75, 76]-=-). Such characterizations can be useful: we often need to exclude certain minors when they are obvious obstructions to some desired property, but knowledge of the structure which their exclusion force... |

48 |
Characterization and recognition of partial 3-trees
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(Show Context)
Citation Context ...c to K5 if and only if it can be obtained from planar graphs and V8 by means of 0-, 1-, 2- and3-sums. There are many similar results in Graph Theory, known as excluded minor theorems (see for example =-=[5, 6, 16, 19, 27, 75, 76]-=-). Such characterizations can be useful: we often need to exclude certain minors when they are obvious obstructions to some desired property, but knowledge of the structure which their exclusion force... |

47 |
Sachs’ linkless embedding conjecture
- Robertson, Seymor, et al.
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(Show Context)
Citation Context ...dding unless it has a minor isomorphic to a member of the Petersen family. It turns out that the related notion of flat embeddings has an interesting theory. The following three results are proved in =-=[50]-=-. Theorem 7.1 A piecewise-linear embedding of a graph G in 3-space is flat if and only if the fundamental group of the complement in 3-space of every subgraph of G is free. Theorem 7.2 Every two flat ... |

40 |
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Citation Context ... surface Σ such that H cannot be drawn in Σ. 6 The graph minor theorem Is there an analogue of Theorem 1.1 for other surfaces? The following is a result of Archdeacon [5], and Glover, Huneke and Wang =-=[14]-=-. Theorem 6.1 AgraphGadmits an embedding in the projective plane if and only if G has no minor isomorphic to a member of an explicit list of 35 graphs. For other surfaces no such theorem is known, and... |

35 |
Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs, Discrete Applied Mathematics 25
- Vazirani, Yannakakis
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(Show Context)
Citation Context ...anents is #P-complete [73] it would seem desirable to have a characterization of matrices for which this is possible. Theorem 11.2 gives such a characterization by a result of Vazirani and Yannakakis =-=[74]-=-. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem [66, 68, 63, 74]. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a ... |

34 |
de Verdière. Sur un nouvel invariant des graphes et un critère de planarité
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Citation Context ...iption of all linklessly embeddable graphs, perhaps along the lines of Theorem 10.3. There is a related result of Lovász and Schrijver [31], concerning the parameter µ introduced by Colin de Verdière =-=[11]-=-. Let G be a connected graph with vertex-set {v1,v2,...,vn}. Then µ(G) is defined as the maximum dimension ofakernelofamatrixM=(mij) n i,j=1 satisfying (i) M is symmetric, (ii) for distinct i, j ∈{1,2... |

34 | Hadwiger’s conjecture for K6-free graphs
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ...’s conjecture for p = 4 is, in fact, equivalent to the 4CT. Robertson, Seymour and the author managed to prove that the next case (that is, p = 5) is also equivalent to the 4CT. More specifically, in =-=[47]-=- they proved the following (without using the 4CT), which immediately implies (assuming the 4CT) Hadwiger’s conjecture for p =5. Wesaythata graph G is apex if G\v is planar for some v ∈ V (G). Theorem... |

32 |
Every planar map is four colorable Part II: reducibility
- Appel, Haken, et al.
- 1977
(Show Context)
Citation Context ...ced, and asked whether the same could be true for any map. Since then the conjecture has attracted a lot of attention and motivated many new developments. A proof was finally found by Appel and Haken =-=[2, 3]-=-, reprinted in [4], formally as follows. Theorem 8.1 Every loopless planar graph is 4-colorable. However, the history seems not to end here. The proof by Appel and Haken is not completely satisfactory... |

31 | Signsolvability revisited - Klee, Ladner, et al. - 1984 |

30 |
Efficiently fourcoloring planar graphs
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- 1996
(Show Context)
Citation Context ...cated that no one has been able to check it. This was partly remedied in a new proof recently found by Robertson, Sanders, Seymour and the author [41], but their proof is still computer-assisted. See =-=[39, 40, 65]-=- for recent surveys. Another aspect of the 4CT is that there are several conjectures that, if true, would generalize the 4CT. It might be possible to reduce some of them to the 4CT, while others may r... |

28 | A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
- Lovász, Schrijver
- 1998
(Show Context)
Citation Context ...dding is flat. It would be nice to have a graph-theoretical description of all linklessly embeddable graphs, perhaps along the lines of Theorem 10.3. There is a related result of Lovász and Schrijver =-=[31]-=-, concerning the parameter µ introduced by Colin de Verdière [11]. Let G be a connected graph with vertex-set {v1,v2,...,vn}. Then µ(G) is defined as the maximum dimension ofakernelofamatrixM=(mij) n ... |

27 |
On chromatic number of finite set-systems
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(Show Context)
Citation Context ...ength. Again,sRecent Excluded Minor Theorems for Graphs 17 Theorem 11.3 provides such an algorithm by [74]. There are other equivalent formulations of the result in terms of 2-coloring of hypergraphs =-=[29, 60]-=-, and several others in terms of sign-nonsingular matrices [9, 26, 67]. Acknowledgements The author acknowledges partial support by NSF under Grant No. DMS9623031, and by NSA under Contract No. MDA904... |

24 |
Tutte, Convex representations of graphs
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(Show Context)
Citation Context ... 4 A splitter theorem for cyclically 5-connected cubic graphs Agraphiscubic if every vertex has degree three. To motivate the next splitter theorem let us mention a special case of a theorem of Tutte =-=[70]-=- (the proof is easy). Theorem 4.1 Let G, H be 3-connected cubic graphs, and let H be a minor of G. Then a graph isomorphic to G can be obtained from H by repeatedly subdividing two distinct edges and ... |

23 |
A characterization of convertible (0,1)-matrices
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(Show Context)
Citation Context ...mputed in polynomial time. Furthermore, the problem of deciding whether a bipartite graph has a Pfaffian orientation is equivalent to several other problems of interest—we mention these later. Little =-=[28]-=- obtained the following “excluded minor” characterization. We say that a graph H is a matching minor ofagraphGif G has a subgraph K such that G\V (K) hasa perfect matching, and H is obtained from K by... |

23 | The four color theorem
- Thomas
- 2007
(Show Context)
Citation Context ...cated that no one has been able to check it. This was partly remedied in a new proof recently found by Robertson, Sanders, Seymour and the author [41], but their proof is still computer-assisted. See =-=[39, 40, 65]-=- for recent surveys. Another aspect of the 4CT is that there are several conjectures that, if true, would generalize the 4CT. It might be possible to reduce some of them to the 4CT, while others may r... |

23 |
Even cycles in directed graphs
- Thomassen
- 1985
(Show Context)
Citation Context ... which this is possible. Theorem 11.2 gives such a characterization by a result of Vazirani and Yannakakis [74]. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem =-=[66, 68, 63, 74]-=-. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a circuit of even length. Again,sRecent Excluded Minor Theorems for Graphs 17 Theorem 11.3 provides such an... |

23 | The even cycle problem for directed graphs
- Thomassen
- 1992
(Show Context)
Citation Context ... which this is possible. Theorem 11.2 gives such a characterization by a result of Vazirani and Yannakakis [74]. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem =-=[66, 68, 63, 74]-=-. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a circuit of even length. Again,sRecent Excluded Minor Theorems for Graphs 17 Theorem 11.3 provides such an... |

21 |
Characterization of even directed graphs
- Seymour, Thomassen
- 1987
(Show Context)
Citation Context ... which this is possible. Theorem 11.2 gives such a characterization by a result of Vazirani and Yannakakis [74]. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem =-=[66, 68, 63, 74]-=-. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a circuit of even length. Again,sRecent Excluded Minor Theorems for Graphs 17 Theorem 11.3 provides such an... |

20 |
On algebraic theory of graph colorings
- Tutte
- 1966
(Show Context)
Citation Context ...alent to the following statement. Theorem 10.1 Every 2-connected cubic planar graph is edge 3-colorable. The smallest 2-connected cubic graph that is not edge 3-colorable is the Petersen graph. Tutte =-=[72]-=- conjectured that Theorem 10.1 holds with “planar” replaced by “no Petersen minor”. Robertson, Sanders, Seymour and the author were recently able to settle Tutte’s conjecture, as follows. Theorem 10.2... |

18 | Sign-nonsingular matrices and even cycles in directed graphs - Thomassen - 1986 |

17 |
Graph Decompositions - a Study in Infinite Graph Theory
- Diestel
- 1990
(Show Context)
Citation Context ...o some desired property, but knowledge of the structure which their exclusion forces may enable us to establish that property for the remaining graphs. Surveys of excluded minor theorems are given in =-=[12]-=- (for finite minors) and [45] (for infinite minors). We show that Theorem 1.1 is not an isolated result, but rather a beginning of a rich theory. We do not attempt to give a complete survey, but inste... |

14 | Linkless Embeddings of Graphs in 3-Space
- Robertson, Thomas
- 1993
(Show Context)
Citation Context ... showed that a graph is planar if and only if µ(G) ≤ 3. This is a surprising result, given the way in which µ is defined. Lovász and Schrijver [31] proved the following generalization, conjectured in =-=[46]-=-. Theorem 7.4 AgraphGhas a linkless embedding if and only if µ(G) ≤ 4. It follows from [11] that this is indeed a generalization of Colin de Verdiére’s result. It is tempting to ask whether there is a... |

13 |
Contractions to K8
- Jørgensen
- 1994
(Show Context)
Citation Context ...apex.sRecent Excluded Minor Theorems for Graphs 14 While Theorem 1.3 gives a structural description of graphs with no K5 minor, Theorem 9.2 does not do the same for graphs with no K6 minor. Jorgensen =-=[22]-=- made the following beautiful conjecture, which implies Theorem 9.2 by a result of Mader [32]. Conjecture 9.3 Every 6-connected graph with no minor isomorphic to K6 is apex. At present, Hadwiger’s con... |

13 |
Nowhere-zero flows, in ``Handbook of Combinatorics
- Seymour
- 1995
(Show Context)
Citation Context ...t no work in that direction has yet been done. Tutte made two other conjectures about nowhere-zero flows, known as the 3-flow [72] and 5flow conjectures [69]. Both of them are still open. We refer to =-=[62]-=- for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I discuss a structural result pertaining to matching theory. An orientation D of a graph G is Pfaffian [23, 24, 30] if every even... |

13 |
Tutte, A theory of 3-connected graphs
- T
- 1961
(Show Context)
Citation Context ...” vertices.) A graph G is k-connected if it has at least k + 1 vertices, and G\X is connected for every set X ⊆ V (G) with |X|<k. (We use \ for deletion.) The following is a classical result of Tutte =-=[71]-=-. Theorem 2.1 Every simple 3-connected graph can be obtained from some wheel by repeatedly applying the operations of adding an edge between two nonadjacent vertices and splitting a vertex. The conver... |

12 |
On a spatial analogue of Kuratowski’s Theorem on planar graphs – an open problem
- Sachs
- 1983
(Show Context)
Citation Context ...lone a polynomial-time one) to decide whether a given graph has a knotless embedding. 7 Linklessly embeddable graphs Related to knotless embeddings are the following two concepts, introduced by Sachs =-=[56, 57]-=- and Böhme [8], respectively. We say that a (piecewise-linear) embedding of a graph in 3-space is linkless if every two disjoint circuits of the graph have zero linking number. We say that an embeddin... |

10 | Matrices of sign-solvable linear systems, Cambridge U - Brualdi, Shader |

9 |
Bemerkungen zu Hadwigers Vermutung
- Wagner
- 1960
(Show Context)
Citation Context ...ollows from Theorem 2.2 as in the proof of Theorem 1.2 that G is isomorphic to V8, as desired. 3 A splitter theorem for internally 4-connected graphs Many excluded minor theorems (e.g. the results of =-=[17, 18, 75, 77]-=-) can be deduced using Theorem 2.2 as in the above proofs of Theorems 1.2 and 1.3. For others, however, it is desirable to have versions of Theorem 2.2 for different kinds of connectivity. Robertson [... |

8 |
Über trennende Eckenmengen in homomorphiekritische Graphen
- Mader
- 1968
(Show Context)
Citation Context ...iption of graphs with no K5 minor, Theorem 9.2 does not do the same for graphs with no K6 minor. Jorgensen [22] made the following beautiful conjecture, which implies Theorem 9.2 by a result of Mader =-=[32]-=-. Conjecture 9.3 Every 6-connected graph with no minor isomorphic to K6 is apex. At present, Hadwiger’s conjecture is open for all p ≥ 6. 10 Tutte’s edge 3-coloring conjecture Tait [64] showed that th... |

8 |
Note on a theorem in geometry of position
- TAIT
(Show Context)
Citation Context ...esult of Mader [32]. Conjecture 9.3 Every 6-connected graph with no minor isomorphic to K6 is apex. At present, Hadwiger’s conjecture is open for all p ≥ 6. 10 Tutte’s edge 3-coloring conjecture Tait =-=[64]-=- showed that the Four Color Theorem is equivalent to the following statement. Theorem 10.1 Every 2-connected cubic planar graph is edge 3-colorable. The smallest 2-connected cubic graph that is not ed... |

7 |
On generating planar graphs
- Barnette
- 1974
(Show Context)
Citation Context ...say that G is a biladder on 2p vertices. We remark that the Petersen graph is a biladder on 10 vertices, and that the dodecahedron is a biladder on 20 vertices. The following theorem [51] generalizes =-=[1, 7, 10, 33, 34]-=-. Figure 2: Biladders Theorem 4.2 Let G, H be cyclically 5-connected cubic graphs, let H be a minor of G, and assume that if H is a biladder, then it is the largest biladder minor of G. Then a graph i... |

7 |
On spatial representations of graphs, in
- Böhme
- 1990
(Show Context)
Citation Context ...ime one) to decide whether a given graph has a knotless embedding. 7 Linklessly embeddable graphs Related to knotless embeddings are the following two concepts, introduced by Sachs [56, 57] and Böhme =-=[8]-=-, respectively. We say that a (piecewise-linear) embedding of a graph in 3-space is linkless if every two disjoint circuits of the graph have zero linking number. We say that an embedding is flat if e... |

7 |
Excluding infinite minors
- Robertson, Seymour, et al.
(Show Context)
Citation Context ...knowledge of the structure which their exclusion forces may enable us to establish that property for the remaining graphs. Surveys of excluded minor theorems are given in [12] (for finite minors) and =-=[45]-=- (for infinite minors). We show that Theorem 1.1 is not an isolated result, but rather a beginning of a rich theory. We do not attempt to give a complete survey, but instead concentrate on the develop... |

6 |
Über einen graphentheoretischen Basisbegri und seine Anwendung aur Färbungsprobleme
- Halin
- 1962
(Show Context)
Citation Context ...c to K5 if and only if it can be obtained from planar graphs and V8 by means of 0-, 1-, 2- and3-sums. There are many similar results in Graph Theory, known as excluded minor theorems (see for example =-=[5, 6, 16, 19, 27, 75, 76]-=-). Such characterizations can be useful: we often need to exclude certain minors when they are obvious obstructions to some desired property, but knowledge of the structure which their exclusion force... |

6 | Kuratowski chains - Robertson, Seymour, et al. - 1995 |

6 | Petersen family minors - Robertson, Seymour, et al. - 1995 |