746:
766:, the cycle with three vertices). This means that a graph is a forest if and only if none of its minors is the loop (or, the cycle with three vertices, respectively). The sole obstruction for the set of paths is the tree with four vertices, one of which has degree 3. In these cases, the obstruction set contains a single element, but in general this is not the case.
203:. For instance, the usual ordering on the non-negative integers is a well-quasi-ordering, but the same ordering on the set of all integers is not, because it contains the infinite descending chain 0, −1, −2, −3... Another example is the set of positive integers ordered by
867:
However, this method requires a specific finite obstruction set to work, and the theorem does not provide one. The theorem proves that such a finite obstruction set exists, and therefore the problem is polynomial because of the above algorithm. However, the algorithm can be used in practice only if
254:
of graphs, there must be a pair of graphs one of which is a minor of the other. The statement that every infinite set has finitely many minimal elements implies this form of the theorem, for if there are only finitely many minimal elements, then each of the remaining graphs must belong to a pair of
214:
The
Robertson–Seymour theorem states that finite undirected graphs and graph minors form a well-quasi-ordering. The graph minor relationship does not contain any infinite descending chain, because each contraction or deletion reduces the number of edges and vertices of the graph (a non-negative
815:
Although the
Robertson–Seymour theorem extends these results to arbitrary minor-closed graph families, it is not a complete substitute for these results, because it does not provide an explicit description of the obstruction set for any family. For example, it tells us that the set of
872:: they prove polynomiality of problems without providing an explicit polynomial-time algorithm. In many specific cases, checking whether a graph is in a given minor-closed family can be done more efficiently: for example, checking whether a graph is planar can be done in linear time.
924:, because of the difficulty of determining the set of forbidden minors. Additionally, the large constant factors involved in these results make them highly impractical. Therefore, the development of explicit fixed-parameter algorithms for these problems, with improved dependence on
255:
this type with one of the minimal elements. And in the other direction, this form of the theorem implies the statement that there can be no infinite antichains, because an infinite antichain is a set that does not contain any pair related by the minor relation.
356:
are closed under taking minors: contracting an edge in a planar graph, or removing edges or vertices from the graph, cannot destroy its planarity. Therefore, the planar graphs have a forbidden minor characterization, which in this case is given by
868:
such a finite obstruction set is provided. As a result, the theorem proves that the problem can be solved in polynomial time, but does not provide a concrete polynomial-time algorithm for solving it. Such proofs of polynomiality are
892:
are minor-closed, the same method applies. For instance, by this result, treewidth, branchwidth, and pathwidth, vertex cover, and the minimum genus of an embedding are all amenable to this approach, and for any fixed
757:
Some examples of finite obstruction sets were already known for specific classes of graphs before the
Robertson–Seymour theorem was proved. For example, the obstruction for the set of all forests is the
402:
The existence of forbidden minor characterizations for all minor-closed graph families is an equivalent way of stating the
Robertson–Seymour theorem. For, suppose that every minor-closed family
820:
has a finite obstruction set, but it does not provide any such set. The complete set of forbidden minors for toroidal graphs remains unknown, but it contains at least 17,535 graphs.
840:
as a minor. This algorithm's running time is cubic (in the size of the graph to check), though with a constant factor that depends superpolynomially on the size of the minor
215:
integer). The nontrivial part of the theorem is that there are no infinite antichains, infinite sets of graphs that are all unrelated to each other by the minor ordering. If
920:
However, this method does not directly provide a single fixed-parameter-tractable algorithm for computing the parameter value for a given graph with unknown
828:
The
Robertson–Seymour theorem has an important consequence in computational complexity, due to the proof by Robertson and Seymour that, for each fixed graph
90:, who proved it in a series of twenty papers spanning over 500 pages from 1983 to 2004. Before its proof, the statement of the theorem was known as
653:
The following sets of finite graphs are minor-closed, and therefore (by the
Robertson–Seymour theorem) have forbidden minor characterizations:
721:
798:} is an obstruction set for the set of all planar graphs, and in fact the unique minimal obstruction set. A similar theorem states that
184:). However, if graphs that are isomorphic may nonetheless be considered as distinct objects, then the minor ordering on graphs forms a
1606:
940:
844:. The running time has been improved to quadratic by Kawarabayashi, Kobayashi, and Reed. As a result, for every minor-closed family
314:
264:
1531:
1501:
1471:
1357:
83:
541:
For the other implication, assume that every set of graphs has a finite subset of minimal graphs and let a minor-closed set
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87:
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on the set of all distinct finite undirected graphs, as it obeys the three axioms of partial orders: it is
1378:
169:
1445:
1415:
657:
274:
243:
forms an antichain; therefore, an equivalent way of stating the theorem is that, in any infinite set
126:
47:. Equivalently, every family of graphs that is closed under minors can be defined by a finite set of
40:
767:
717:
358:
192:
157:
102:
52:
44:
1330:
710:
706:
153:
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110:
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of a graph is the total number of its vertices and edges, and ≤ denotes the minor ordering.)
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1401:
1339:
1325:
1294:
1279:
948:
869:
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32:
1353:
1321:
1293:, Handbooks in Operations Research and Management Science, vol. 7, pp. 481–502,
881:
833:
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695:
228:
48:
905:. A problem with this property, that it can be solved in polynomial time for any fixed
817:
691:
661:
366:
114:
60:
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897:
there is a polynomial time algorithm for testing whether these invariants are at most
1590:
1492:
1441:
149:
36:
385:, and the planar graphs are exactly the graphs that do not have a minor in the set {
679:
673:
669:
353:
208:
204:
56:
20:
1432:
1386:
247:
of graphs, there must be only a finite number of non-isomorphic minimal elements.
188:, a relation that is reflexive and transitive but not necessarily antisymmetric.
1364:(1987), "The metamathematics of the graph minor theorem", in Simpson, S. (ed.),
901:, in which the exponent in the running time of the algorithm does not depend on
733:
1552:
1406:
745:
687:
665:
1578:
848:, there is polynomial time algorithm for testing whether a graph belongs to
729:
725:
200:
1522:
1328:(1988), "Nonconstructive tools for proving polynomial-time decidability",
699:
305:). According to the Robertson–Seymour theorem, there exists a finite set
185:
1344:
250:
Another equivalent form of the theorem is that, in any infinite set
365:
of minor-minimal nonplanar graphs contains exactly two graphs, the
1462:
744:
277:
under the operation of taking minors if every minor of a graph in
227:
containing one representative graph for each equivalence class of
1446:"A Large Set of Torus Obstructions and How They Were Discovered"
1315:(Electronic Edition 2005 ed.), Springer, pp. 326–367
956:
812:
are the forbidden minors for the set of outerplanar graphs.
770:
states that a graph is planar if and only if it has neither
82:
The
Robertson–Seymour theorem is named after mathematicians
1508:(1995), "Graph Minors. XIII. The disjoint paths problem",
1227:
207:, which has no infinite descending chains, but where the
936:
140:
by a sequence of zero or more contractions of edges of
947:
in various formal systems that are much stronger than
690:(formed by adding a single vertex to a planar graph),
1284:"Algorithmic implications of the graph minor theorem"
928:, has continued to be an important line of research.
325:
are exactly the graphs that do not have any graph in
1306:
Diestel, Reinhard (2005), "Minors, Trees, and WQO",
720:
of size bounded by some fixed constant; graphs with
422:
as the family of graphs that do not have a minor in
561:be the graphs which are not minors of any graph in
1538:(2004), "Graph Minors. XX. Wagner's conjecture",
1262:
1242:
1184:
1145:
1071:
1238:
1100:
1096:
1083:
852:: simply check whether the given graph contains
98:, although Wagner said he never conjectured it.
1478:(1983), "Graph Minors. I. Excluding a forest",
1258:
1215:
939:showed that the following theorem exhibits the
1387:"The disjoint paths problem in quadratic time"
1420:Bulletin of the American Mathematical Society
753:, the obstruction set for linkless embedding.
8:
1199:
1197:
724:bounded by some fixed constant; graphs with
180:are minors of each other, then they must be
16:Finiteness of sets of forbidden graph minors
553:if and only if it does not have a minor in
1368:, Contemporary Mathematics, vol. 65,
1228:Kawarabayashi, Kobayashi & Reed (2012)
990:is a sequence of finite undirected graphs,
836:algorithm for testing whether a graph has
709:in Euclidean 3-space, and graphs that are
1551:
1540:Journal of Combinatorial Theory, Series B
1521:
1510:Journal of Combinatorial Theory, Series B
1491:
1480:Journal of Combinatorial Theory, Series B
1461:
1431:
1405:
1394:Journal of Combinatorial Theory, Series B
1343:
909:with an exponent that does not depend on
534:is the finite set of minimal elements of
235:but for which no proper minor belongs to
59:as being the graphs that do not have the
1168:
1166:
937:Friedman, Robertson & Seymour (1987)
1450:The Electronic Journal of Combinatorics
1157:
1128:
1116:
1104:
1064:
569:be the finite set of minimal graphs in
293:be the class of graphs that are not in
144:and deletions of edges and vertices of
136:is any graph that may be obtained from
1246:
1203:
1188:
1172:
1132:
932:Finite form of the graph minor theorem
414:be any infinite set of graphs. Define
1245:, Theorem 2.1.4 and Corollary 2.1.5;
784:as a minor. In other words, the set {
430:is minor-closed and has a finite set
410:of minimal forbidden minors, and let
7:
1095:Robertson and Seymour (
888:, the graphs with invariant at most
156:(every graph is a minor of itself),
113:and proved in 1960 independently by
289:is a minor-closed family, then let
109:, which was conjectured in 1937 by
549:of graphs such that a graph is in
211:constitute an infinite antichain.
14:
1148:, Section 2, "well-quasi-orders".
884:with the property that, for each
722:Colin de Verdière graph invariant
649:Examples of minor-closed families
526:have a minor among the graphs in
434:of minimal forbidden minors. Let
259:Forbidden minor characterizations
148:. The minor relationship forms a
713:embeddable in Euclidean 3-space;
545:be given. We want to find a set
315:forbidden graph characterization
313:. These minimal elements form a
265:Forbidden graph characterization
1263:Bienstock & Langston (1995)
1243:Bienstock & Langston (1995)
1185:Bienstock & Langston (1995)
1146:Bienstock & Langston (1995)
1072:Bienstock & Langston (1995)
762:graph (or, if one restricts to
736:bounded by some fixed constant.
621:is not a minor of any graph in
94:after the German mathematician
1418:(2005), "Graph Minor Theory",
1239:Robertson & Seymour (1995)
1084:Robertson & Seymour (2004)
573:. Now, let an arbitrary graph
478:cannot have a proper minor in
1:
1444:; Woodcock, Jennifer (2018),
1433:10.1090/S0273-0979-05-01088-8
1370:American Mathematical Society
1299:10.1016/S0927-0507(05)80125-2
1259:Fellows & Langston (1988)
1216:Myrvold & Woodcock (2018)
966:: For every positive integer
698:on any fixed two-dimensional
694:, and the graphs that can be
191:A preorder is said to form a
1493:10.1016/0095-8956(83)90079-5
955:in systems much weaker than
876:Fixed-parameter tractability
629:is minor-closed. Therefore,
577:be given. Assume first that
1574:"Robertson-Seymour Theorem"
824:Polynomial time recognition
329:as a minor. The members of
1623:
1553:10.1016/j.jctb.2004.08.001
1407:10.1016/j.jctb.2011.07.004
522:, and all other graphs in
462:are the minimal graphs in
343:minor-minimal obstructions
262:
195:if it contains neither an
1135:, Section 3.3, pp. 78–79.
915:fixed-parameter tractable
856:for each forbidden minor
197:infinite descending chain
25:Robertson–Seymour theorem
1607:Theorems in graph theory
1191:, Theorem 4, p. 78.
377:complete bipartite graph
273:of graphs is said to be
219:is a set of graphs, and
71:complete bipartite graph
1379:Kawarabayashi, Ken-ichi
1366:Logic and Combinatorics
1054:Graph structure theorem
589:cannot have a minor in
502:would be an element in
309:of minimal elements in
231:(graphs that belong to
160:(a minor of a minor of
51:, in the same way that
1523:10.1006/jctb.1995.1006
970:, there is an integer
754:
107:Kruskal's tree theorem
1381:; Kobayashi, Yusuke;
1131:, pp. 335–336);
864:’s obstruction set.
748:
707:linklessly embeddable
494:must have a minor in
438:be the complement of
164:is itself a minor of
43:relationship, form a
1326:Langston, Michael A.
1280:Langston, Michael A.
1249:, Theorem 11, p. 83.
943:phenomenon by being
490:. At the same time,
101:A weaker result for
1345:10.1145/44483.44491
1322:Fellows, Michael R.
1278:Bienstock, Daniel;
1187:, Corollary 2.1.1;
718:feedback vertex set
466:. Consider a graph
193:well-quasi-ordering
92:Wagner's conjecture
45:well-quasi-ordering
29:graph minor theorem
1597:Graph minor theory
1570:Weisstein, Eric W.
1372:, pp. 229–261
1331:Journal of the ACM
1206:, pp. 76–77).
755:
684:outerplanar graphs
660:, linear forests (
609:. Now assume that
498:, since otherwise
458:are disjoint, and
117:and S. Tarkowski.
55:characterizes the
31:) states that the
1002:has size at most
974:so large that if
510:is an element in
406:has a finite set
352:For example, the
345:) for the family
37:partially ordered
33:undirected graphs
27:(also called the
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870:non-constructive
768:Wagner's theorem
741:Obstruction sets
705:graphs that are
359:Wagner's theorem
339:forbidden minors
321:: the graphs in
281:also belongs to
229:minimal elements
199:nor an infinite
131:undirected graph
53:Wagner's theorem
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751:Petersen family
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662:disjoint unions
651:
641:has a minor in
605:is a subset of
518:is a subset of
446:is a subset of
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335:excluded minors
333:are called the
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172:(if two graphs
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111:Andrew Vázsonyi
88:Paul D. Seymour
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1422:, New Series,
1416:Lovász, László
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1426:(1): 75–86,
1423:
1419:
1397:
1393:
1365:
1335:
1329:
1309:Graph Theory
1308:
1290:
1265:, Section 6.
1254:
1234:
1223:
1211:
1204:Lovász (2005
1180:
1173:Lovász (2005
1153:
1140:
1124:
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944:
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935:
925:
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251:
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236:
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224:
220:
216:
213:
205:divisibility
190:
177:
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145:
141:
137:
133:
124:
100:
96:Klaus Wagner
91:
81:
73:
63:
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21:graph theory
18:
1383:Reed, Bruce
1040:(Here, the
993:where each
734:branchwidth
688:apex graphs
666:path graphs
79:as minors.
41:graph minor
1591:Categories
1272:References
1144:E.g., see
945:unprovable
711:knotlessly
613:is not in
565:, and let
361:: the set
299:complement
182:isomorphic
158:transitive
1579:MathWorld
1028:for some
730:pathwidth
726:treewidth
269:A family
201:antichain
154:reflexive
121:Statement
1385:(2012),
1282:(1995),
1048:See also
953:provable
700:manifold
696:embedded
625:, since
514:, i.e.,
375:and the
239:), then
186:preorder
1010:, then
981:, ...,
964:Theorem
791:,
658:forests
617:. Then
426:. Then
392:,
168:), and
69:or the
39:by the
672:, and
633:is in
597:is in
593:since
581:is in
557:. Let
482:since
450:since
275:closed
129:of an
23:, the
1390:(PDF)
1313:(PDF)
1287:(PDF)
1060:Notes
1032:<
732:, or
637:, so
530:, so
418:from
341:, or
297:(the
285:. If
127:minor
103:trees
1101:2004
1097:1983
1042:size
880:For
805:and
777:nor
760:loop
749:The
601:and
454:and
337:(or
176:and
86:and
1548:doi
1518:doi
1488:doi
1458:doi
1428:doi
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957:ZFC
860:in
810:2,3
796:3,3
782:3,3
668:),
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470:in
399:}.
397:3,3
383:3,3
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301:of
77:3,3
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390:5
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241:M
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146:G
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74:K
67:5
64:K
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