## Recent Excluded Minor Theorems

Venue: | SURVEYS IN COMBINATORICS, LMS LECTURE NOTE SERIES |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Thomas_recentexcluded,

author = {Robin Thomas},

title = {Recent Excluded Minor Theorems},

booktitle = {SURVEYS IN COMBINATORICS, LMS LECTURE NOTE SERIES},

year = {},

publisher = {Cambridge University Press}

}

### OpenURL

### Abstract

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

### Citations

529 |
Matching Theory
- Lovász, Plummer
- 1986
(Show Context)
Citation Context ...efer to [61] for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I would like to discuss a structural result pertaining to matching theory. An orientation D of a graph G is Pfaffian =-=[22, 23, 29]-=- if every even circuit C of G such that GnV (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 ... |

286 |
Every planar map is four colorable
- Appel, Haken
- 1976
(Show Context)
Citation Context ...ther the same could be true for any map. Since then the conjecture attracted a lot of attention, and motivated many new developments. A proof was finally found by Appel and Haken [1, 2], reprinted in =-=[3]-=-, formally as follows. Theorem 8.1 Every loopless planar graph is 4-colorable. However, the history does not seem to end here. The proof by Appel and Haken is not completely satisfactory, because it r... |

255 |
Graph minors XIII: the disjoint paths problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ....1. That structure can be expoited to conclude the proof, but the argument is lengthy and depends on the results of several other papers. The following is another deep result of Robertson and Seymour =-=[41]-=-. Theorem 6.3 For every graph H there exists an O(n 3 ) algorithm to decide whether an input graph on n vertices has a minor isomorphic to H. Theorems 6.2 and 6.3 have some surprising consequences. Co... |

170 |
Sur le probleme des courbes gauches en topologie
- Kuratowski
- 1930
(Show Context)
Citation Context ...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 [74] of Kuratowski's theorem =-=[26]-=-. Theorem 1.1 A graph is planar if and only if it has no subgraph isomorphic to a minor of K 5 or K 3;3 . Kuratowski's theorem is important, because it gives a good characterization (in the sense of E... |

165 |
Decomposition of regular matroids
- Seymour
- 1980
(Show Context)
Citation Context .... Notice that the converse also holds: if a graph can be obtained from a wheel (in fact, any simple 3-connected graph) by means of the above two operations, then it is simple and 3-connected. Seymour =-=[60]-=- proved the following strengthening. Theorem 2.2 Let H be a simple 3-connected minor of a simple 3-connected graph G such that if H is a wheel, then H is the largest wheel minor of G. Then a graph iso... |

117 | The four colour theorem
- Robertson, Sanders, et al.
- 1997
(Show Context)
Citation Context ...omputers, and even the computer-free part is so complicated 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 =-=[40]-=-, but their proof is still computer-assisted. See [38, 39, 64] for recent surveys. Another aspect of the 4CT is that there are several conjectures that, if true, would generalize the 4CT. It might be ... |

86 |
Dimer statistics and phase transitions
- Kasteleyn
- 1963
(Show Context)
Citation Context ...efer to [61] for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I would like to discuss a structural result pertaining to matching theory. An orientation D of a graph G is Pfaffian =-=[22, 23, 29]-=- if every even circuit C of G such that GnV (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 ... |

66 |
Über eine Klassifikation der Streckenkomplexe, Vierteljschr
- Hadwiger
- 1943
(Show Context)
Citation Context ...f them to the 4CT, while others may require a strengthening of the proof of the Four Color Theorem. We will discuss two such generalizations in the next two sections. 9 Hadwiger's conjecture Hadwiger =-=[14]-=- made the following conjecture. Conjecture 9.1 For every integer ps1, every loopless graph with no K p+1 minor is p-colorable. Conjecture 9.1 is trivial for ps2, for p = 3 it was shown by Hadwiger [14... |

64 |
The complexity of computing the permanent, Theoret
- Valiant
- 1979
(Show Context)
Citation Context ...a1's in such a way that the determinant of B equals the permanent of A. This cannot be done for all matrices. However, Recent Excluded Minor Theorems 16 given that computing permanents is #P-complete =-=[72]-=- 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 [73]. Another consequence ... |

58 |
Tutte, ‘A contribution to the theory of chromatic polynomials
- T
- 1954
(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 [71] and 5flow conjectures =-=[68]-=-. Both of them are still open. We refer to [61] for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I would like to discuss a structural result pertaining to matching theory. An orie... |

57 | pfaffian orientations, and even directed circuits
- Robertson, Seymour, et al.
- 1999
(Show Context)
Citation Context ... seem to imply a polynomial-time algorithm to test if a given bipartite graph has a Pfaffian orientation. The next theorem, proven independently McCuaig [34, 35], and by Robertson, Seymour and Thomas =-=[53]-=-, 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... |

53 |
Graph theory and crystal physics,” in Graph theory and theoretical physics , edited by F
- Kasteleyn
- 1967
(Show Context)
Citation Context ...efer to [61] for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I would like to discuss a structural result pertaining to matching theory. An orientation D of a graph G is Pfaffian =-=[22, 23, 29]-=- if every even circuit C of G such that GnV (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 ... |

50 |
A Kuratowski theorem for the projective plane
- Archdeacon
- 1981
(Show Context)
Citation Context ...y if it can be obtained from planar graphs and V 8 by means of 0-, 1-, 2- and 3-sums. There is a large collection of similar results in Graph Theory, known as excluded minor theorems (see for example =-=[4, 5, 15, 18, 26, 74, 75]-=-). Such characterizations can be useful: we often need to exclude certain minors when Recent Excluded Minor Theorems 2 they are obvious obstructions to some desired property, but knowledge of the stru... |

50 |
Characterization and recognition of partial 3-trees
- Arnborg, Proskurowski
- 1986
(Show Context)
Citation Context ...y if it can be obtained from planar graphs and V 8 by means of 0-, 1-, 2- and 3-sums. There is a large collection of similar results in Graph Theory, known as excluded minor theorems (see for example =-=[4, 5, 15, 18, 26, 74, 75]-=-). Such characterizations can be useful: we often need to exclude certain minors when Recent Excluded Minor Theorems 2 they are obvious obstructions to some desired property, but knowledge of the stru... |

49 |
Über eine Eigenschaft der ebenen
- Wagner
- 1937
(Show Context)
Citation Context ...el edges. A graph is a minor 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 =-=[74]-=- of Kuratowski's theorem [26]. Theorem 1.1 A graph is planar if and only if it has no subgraph isomorphic to a minor of K 5 or K 3;3 . Kuratowski's theorem is important, because it gives a good charac... |

46 |
Sachs’ linkless embedding conjecture
- Robertson, Seymor, et al.
- 1995
(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 =-=[49]-=-. 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. Recent Excluded Minor Theor... |

40 |
103 graphs that are irreducible for the projective plane
- Glover, Huneke, et al.
- 1979
(Show Context)
Citation Context ...Sigma such that H cannot be drawn in \Sigma. 6 The graph minor theorem Is there an analogue of Theorem 1.1 for other surfaces? The following is a result of Archdeacon [4], and Glover, Huneke and Wang =-=[13]-=-. Theorem 6.1 A graph G admits 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, ... |

36 |
Every planar map is four colorable, Part I: discharging
- Appel, Haken
(Show Context)
Citation Context ...fficed, and asked whether the same could be true for any map. Since then the conjecture attracted a lot of attention, and motivated many new developments. A proof was finally found by Appel and Haken =-=[1, 2]-=-, reprinted in [3], formally as follows. Theorem 8.1 Every loopless planar graph is 4-colorable. However, the history does not seem to end here. The proof by Appel and Haken is not completely satisfac... |

35 |
Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs, Discrete Applied Mathematics 25
- Vazirani, Yannakakis
- 1989
(Show Context)
Citation Context ...anents is #P-complete [72] 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 =-=[73]-=-. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem [65, 67, 62, 73]. 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
- Colin
- 1990
(Show Context)
Citation Context ... = (x ij ) n i;j=1 such that MX = 0 and such that x ij = 0 whenever i = j or m ij 6= 0. If G is not connected we define ��(G) to be the maximum of ��(H) over all components H of G. Colin de Ve=-=rdi`ere [10] showed -=-that a graph is planar if and only if ��(G)s3. This is a surprising result, given the way �� is defined. Lov'asz and Schrijver [30] proved the following generalization, conjectured in [45]. Th... |

32 |
Every planar map is four colorable, Part II: reducibility
- Appel, Haken, et al.
(Show Context)
Citation Context ...fficed, and asked whether the same could be true for any map. Since then the conjecture attracted a lot of attention, and motivated many new developments. A proof was finally found by Appel and Haken =-=[1, 2]-=-, reprinted in [3], formally as follows. Theorem 8.1 Every loopless planar graph is 4-colorable. However, the history does not seem to end here. The proof by Appel and Haken is not completely satisfac... |

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

31 |
Efficiently Four-Coloring Planar Graphs
- Robertson, Sanders, et al.
- 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 [40], but their proof is still computer-assisted. See =-=[38, 39, 64]-=- 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 |
On chromatic number of finite set-systems
- Lovász
- 1968
(Show Context)
Citation Context ...o decide if a digraph has a circuit of even length. Again, Theorem 11.3 provides such an algorithm by [73]. There are other equivalent formulations of the result in terms of 2-coloring of hypergraphs =-=[28, 59]-=-, and several others in terms of sign-nonsingular matrices [8, 25, 66]. Acknowledgements The author would like to acknowledge partial support by NSF under Grant No. DMS-9623031, and by NSA under Contr... |

28 | A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
- Lovász, Schrijver
- 1998
(Show Context)
Citation Context ...imum of ��(H) over all components H of G. Colin de Verdi`ere [10] showed that a graph is planar if and only if ��(G)s3. This is a surprising result, given the way �� is defined. Lov'asz an=-=d Schrijver [30] pro-=-ved the following generalization, conjectured in [45]. Theorem 7.4 A graph G has a linkless embedding if and only if ��(G)s4. It follows from [10] that this is indeed a generalization of Colin de ... |

25 |
Tutte, Convex Representations of Graphs
- T
- 1960
(Show Context)
Citation Context ...A splitter theorem for cyclically 5-connected cubic graphs A graph is cubic if every vertex has degree three. To motivate the next splitter theorem let us mention a special case of a theorem of Tutte =-=[69]-=- (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 | An update on the four-color theorem
- Thomas
- 1998
(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 [40], but their proof is still computer-assisted. See =-=[38, 39, 64]-=- 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 [73]. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem =-=[65, 67, 62, 73]-=-. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a circuit of even length. Again, Theorem 11.3 provides such an algorithm by [73]. There are other equivalen... |

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 [73]. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem =-=[65, 67, 62, 73]-=-. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a circuit of even length. Again, Theorem 11.3 provides such an algorithm by [73]. There are other equivalen... |

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 [73]. Another consequence of Theorem 11.2 is a solution of the even directed circuit problem =-=[65, 67, 62, 73]-=-. The question is whether there exists a polynomialtime algorithm to decide if a digraph has a circuit of even length. Again, Theorem 11.3 provides such an algorithm by [73]. There are other equivalen... |

21 |
On the 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 =-=[71] conjectured that Th-=-eorem 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 |
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 =-=[11]-=- (for finite minors) and [44] (for infinite minors). We shall see that Theorem 1.1 is not an isolated result, but rather a beginning of a rich theory. We shall not attempt to give a complete survey, b... |

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

17 |
Note on a theorem in the geometry of position, Trans
- Tait
(Show Context)
Citation Context ... result of Mader [31]. Conjecture 9.3 Every 6-connected graph with no minor isomorphic to K 6 is apex. Presently, Hadwiger's conjecture is open for all ps6. 10 Tutte's edge 3-coloring conjecture Tait =-=[63]-=- has shown 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... |

16 | Linkless embeddings of graphs in 3-space
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ...ere [10] showed that a graph is planar if and only if ��(G)s3. This is a surprising result, given the way �� is defined. Lov'asz and Schrijver [30] proved the following generalization, conject=-=ured in [45]. Th-=-eorem 7.4 A graph G has a linkless embedding if and only if ��(G)s4. It follows from [10] that this is indeed a generalization of Colin de Verdi'ere's result. It is tempting to ask whether there i... |

15 |
On the two-colouring of hypergraphs, Quart
- Seymour
- 1974
(Show Context)
Citation Context ...o decide if a digraph has a circuit of even length. Again, Theorem 11.3 provides such an algorithm by [73]. There are other equivalent formulations of the result in terms of 2-coloring of hypergraphs =-=[28, 59]-=-, and several others in terms of sign-nonsingular matrices [8, 25, 66]. Acknowledgements The author would like to acknowledge partial support by NSF under Grant No. DMS-9623031, and by NSA under Contr... |

14 |
On spatial analogue of Kuratowski’s theorem on planar graphs
- 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 =-=[55, 56]-=- and Bohme [7], 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... |

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 [71] and 5flow conjectures [68]. Both of them are still open. We refer to =-=[61]-=- for a survey on nowhere-zero flows. 11 Pfaffian orientations Finally, I would like to discuss a structural result pertaining to matching theory. An orientation D of a graph G is Pfaffian [22, 23, 29]... |

13 |
Tutte, A theory of 3-connected graphs
- T
- 1961
(Show Context)
Citation Context ...ined from a circuit on at least three vertices by adding a vertex joined to every vertex on the circuit. (Paths and circuits have no "repeated" vertices.) The following is a classical result=-= of Tutte [70]-=-. Theorem 2.1 Every simple 3-connected graph can be obtained from a wheel by repeatedly applying the operations of adding an edge between two nonadjacent vertices and splitting a vertex. Notice that t... |

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

10 |
Homomorphism theorems for graphs
- Dirac
- 1964
(Show Context)
Citation Context ...llowing conjecture. Conjecture 9.1 For every integer ps1, every loopless graph with no K p+1 minor is p-colorable. Conjecture 9.1 is trivial for ps2, for p = 3 it was shown by Hadwiger [14] and Dirac =-=[12]-=- (the proof is not very difficult), but for ps4 it seems very difficult, because it implies the Four Color Theorem. To see that let ps4, and let G be a planar graph. Let H be obtained from G by adding... |

9 |
Bemerkungen zu Hadwigers Vermutung
- Wagner
- 1960
(Show Context)
Citation Context ...heorem 2.2 similarly as in the proof of Theorem 1.2 that G is isomorphic to V 8 , as desired. 3 A splitter theorem for internally 4-connected graphs Many excluded minor theorems (e.g., the results of =-=[16, 17, 74, 76]-=-) can be deduced using Theorem 2.2 similarly 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. R... |

8 |
Über trennende Eckenmengen in homomorphiekritische Graphen
- Mader
- 1968
(Show Context)
Citation Context ...tion of graphs with no K 5 minor, Theorem 9.2 does not do the same for graphs with no K 6 minor. Jorgensen [21] made the following beautiful conjecture, which implies Theorem 9.2 by a result of Mader =-=[31]-=-. Conjecture 9.3 Every 6-connected graph with no minor isomorphic to K 6 is apex. Presently, Hadwiger's conjecture is open for all ps6. 10 Tutte's edge 3-coloring conjecture Tait [63] has shown that t... |

8 |
A new proof of the Four-Color Theorem
- Robertson, Sanders, et al.
- 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 [40], but their proof is still computer-assisted. See =-=[38, 39, 64]-=- 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... |

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 [50] generalizes =-=[6, 9, 32, 33]-=-. 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 |
A characterization of convertible (0
- Little
- 1975
(Show Context)
Citation Context ...n polynomial time. Furthermore, the problem of deciding whether a bipartite graph has a Pfaffian orientation is equivalent to several other problems of interest --- we will mention that later. Little =-=[27] obtained -=-the following "excluded minor" characterization. We say that a graph H is a matching minor of a graph G if G has a subgraph K such that GnV (K) has a perfect matching, and H is obtained from... |

7 |
Excluding infinite minors
- Robertson, Seymour, et al.
- 1991
(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 [11] (for finite minors) and =-=[44]-=- (for infinite minors). We shall see that Theorem 1.1 is not an isolated result, but rather a beginning of a rich theory. We shall not attempt to give a complete survey, but instead will concentrate o... |

6 |
On spatial representations of graphs, Contemporary Methods in Graph Theory
- 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 [55, 56] and Bohme =-=[7]-=-, 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... |

6 |
Über einen graphentheoretischen Basisbegri und seine Anwendung aur Färbungsprobleme
- Halin
- 1962
(Show Context)
Citation Context ...y if it can be obtained from planar graphs and V 8 by means of 0-, 1-, 2- and 3-sums. There is a large collection of similar results in Graph Theory, known as excluded minor theorems (see for example =-=[4, 5, 15, 18, 26, 74, 75]-=-). Such characterizations can be useful: we often need to exclude certain minors when Recent Excluded Minor Theorems 2 they are obvious obstructions to some desired property, but knowledge of the stru... |

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