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....Conjecture TEH Yee Whye 009753429 January 19, 2000 Abstract The study of graph minors is an important area of topological graph theory and graph embeddings. Central to the study of graph minors is Wagner s conjecture, which was proven by Robertson and Seymour [15]. We survey the ideas and results leading up to the proof of Wagner s conjecture and their implications. Contents 1 Introduction 2 2 Preliminaries, definitions and notations 3 3 Well quasi orderings 4 4 Trees 5 5 Tree decompositions 6 6 Wagner s conjecture 9 7 The disjoint paths problem 11 8 ....

....F S are called the excluded minors of embeddability in S. In the same year Bodendiek and Wagner [4] showed a similar result for the orientable surfaces. Robertson and Seymour took a less direct but more fruitful approach to the problem. In a long series of papers they proved Wagner s conjecture [15] : Theorem 1.1 (Wagner s conjecture) If G 1 ; G 2 ; G 3 ; is a sequence of graphs there exists 1 i j such that G i is isomorphic to a minor of G j . Now consider the family F of graphs not embeddable in some given surface S. Let F S ae F be those members in F that are minor minimal ....

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N. Robertson and P. D. Seymour. Graph minors XX : Wagner's conjecture. Manuscript.

....of rooted (di)graphs. Our new result in Chapter 2 is obtained by an application of a consequence of the theory of line graphs [23, 37, 44] Moreover, it concludes that the Minor Theorem for antimatroids does not hold, while the Graph Minor Theorem is shown to hold by Robertson Seymour [39]. Secondly, in Chapter 4 we consider an antimatroidal analogue of Dilworth s decomposition theorem for partially ordered sets. In a paper of 1950, Dilworth [9] showed one of the most famous theorems for partially ordered sets, or posets. That is, for any poset, the maximum size of antichains is ....

....and Q = rq 1 Delta Delta Delta q n z for l; m; n 0. If we delete p and x, and shrink y 1 , y l , w 1 , wm , q 1 , q n to r, then it is 27 reduced to A or B. Case 4. jT j = 4. It is easily checked that this case is reduced to Case 3 1 or Case 3 3. Robertson Seymour [39] have shown the Graph Minor Theorem, that is, in every infinite set of graphs there are two graphs such that one is a minor of the other. From this theorem, we conclude that every minor closed property of graphs can be characterized by finitely many forbidden minors. But for antimatroids, Theorem ....

N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture.

....of line search antimatroids of rooted digraphs as below. Corollary 4. Let F be an antimatroid. Then, F is a line search antimatroid of a rooted digraph if and only if F has no minor isomorphic to D 5 or the point search antimatroids of A, B, C m;n or D l;m;n (l; m;n 1) Robertson Seymour [12] have shown the Graph Minor Theorem, that is, in every infinite set of graphs there are two graphs such that one is a minor of the other. From this theorem, we conclude that every minor closed property of graphs can be characterized by finitely many forbidden minors. But for antimatroids, Theorem ....

N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture.

....graphs are called NOC graphs. The property of being outer cylindrical is preserved under edge deletion, edge contraction, and deleting isolated vertices. In other words, the class of outer cylindrical graphs is closed under the taking of minors. It follows from Robertson and Seymour s [12] proof of Wagner s conjecture that there is a nite obstruction set to being outer cylindrical; that is, a graph is outercylindrical if and only if it does not contain any of a nite number of graphs as a minor. We nd this set. Theorem 1.1 The Main Result: There are exactly 38 minor minimal ....

N. Robertson and P.D. Seymour, Graph Minors XX: Wagner's conjecture, manuscript. 28

....of rooted (di)graphs. Our new result in Chapter 2 is obtained by an application of a consequence of the theory of line graphs [23, 37, 44] Moreover, it concludes that the Minor Theorem for antimatroids does not hold, while the Graph Minor Theorem is shown to hold by Robertson Seymour [39]. Secondly, in Chapter 4 we consider an antimatroidal analogue of Dilworth s decomposition theorem for partially ordered sets. In a paper of 1950, Dilworth [9] showed one of the most famous theorems for partially ordered sets, or posets. That is, for any poset, the maximum size of antichains is ....

....qw and Q = rq 1 Delta Delta Delta q n z for l; m; n 0. If we delete p and x, and shrink y 1 , y l , w 1 , wm , q 1 , q n to r, then it is 27 reduced to A or B. Case 4. jT j = 4. It is easily checked that this case is reduced to Case 3 1 or Case 3 3. Robertson Seymour [39] have shown the Graph Minor Theorem, that is, in every infinite set of graphs there are two graphs such that one is a minor of the other. From this theorem, we conclude that every minor closed property of graphs can be characterized by finitely many forbidden minors. But for antimatroids, Theorem ....

N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture.

....graphs are called NOC graphs. The property of being outer cylindrical is preserved under edge deletion, edge contraction, and deleting isolated vertices. In other words, the class of outer cylindrical graphs is closed under the taking of minors. It follows from Robertson and Seymour s [12] proof of Wagner s conjecture that there is a finite obstruction set to being outer cylindrical; that is, a graph is outercylindrical if and only if it does not contain any of a finite number of graphs as a minor. We find this set. Theorem 1.1 The Main Result: There are exactly 38 minor minimal ....

N. Robertson and P.D. Seymour, Graph Minors XX: Wagner's conjecture, manuscript.

....graphs were originally found by Glover, Huneke, and Wang [Wan,GHW] their list was proven complete by Archdeacon [A1,A2] There are 103 such graphs. Mahader [M] showed that exactly 35 of these graphs are minor minimal; this work is also implicit in [A1,A2] Robertson and Seymour [RS3] have proven Wagner s Conjecture: that there does not exist an infinite set of graphs which are pairwise noncomparable under the minor order. It follows that for any hereditary property the set of obstructions for the minor order is finite. In particular, the OBSTRUCTION SETS FOR ....

N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture, (preprint) (1988).

....de ned as a set of abstract graphs : its elements are by de nition pairwise nonisomorphic. This set is called the set of obstructions of C. 7.1. 2 Fact : For every minor closed class C of graphs : C = FORB(OBST (C) The major result in this eld is the following result by Robertson and Seymour [56]. Theorem 7.1 : Graph Minor Theorem) Every in nite set of undirected graphs contains two graphs comparable by . Hence for every class C of graphs, the set OBST (C) is nite. The obstructions of minor closed classes are known in relatively few cases. We have already mentioned planar graphs. The ....

: ROBERTSON N., SEYMOUR P., Graph minors XX : Wagner's conjecture, September 1988.

....by a finite set of forbidden minors. In other words, the set of minor minimal graphs not in the class must be finite. For example, the class of series parallel graphs is minor closed and is characterized by the exclusion of a K 4 minor. Wagner s conjecture was proved by Robertson and Seymour [20] in a remarkable sequence of papers. Truemper [21] showed that the class of Y DeltaY reducible graphs is minor closed (he assumed that the minor was 2 connected, since he did not allow degree one reductions) From the above it follows that there are a finite number of minor minimal graphs which ....

N. Robertson and P.D. Seymour. Graph minors XX: Wagner's Conjecture. preprint, 1988.

.... planar graphs as precisely those which do not contain K 5 or K 3;3 as a minor; the exclusion of minors has since been used as a nice way to extend results for planar graphs to more general classes [2, 30] and as the basis for some very deep results in both structural and algorithmic graph theory [31, 32]. We show the following. For every ff 0 there is a number 0 such that the following holds: Every bounded degree ff expander graph G on n nodes contains every graph H with O(n= log n) nodes and edges as a minor. Moreover, we give a randomized polynomial time algorithm, similar to that of ....

N. Robertson, P.D. Seymour, "Graph Minors XX. Wagner's conjecture," submitted for publication.

....and only if, for every H m G, H is not in Q. If F is a minor closed family, then the obstruction set for F , written obs(F ) is the set of all minimal graphs in the complement of F . Therefore, if F is a minor closed family, then G 2 F if and only if H 6 m G for every H 2 obs(F ) Theorem 4. 3 [RS4] If F is a minor closed family of finite graphs, then obs(F) is finite. Theorem 4.4 [RS3] For every fixed graph H, there exists a polynomial time algorithm that, when given an input graph G, decides whether H is a minor of G. Thus there is a polynomial time algorithm to decide membership for any ....

N. Robertson and P. D. Seymour, "Graph Minors XX. Wagner's Conjecture, " manuscript (1988).

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N. Robertson and P. Seymour, Graph minors XX: Wagner's conjecture, manuscript.

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N. Robertson and P. Seymour, Graph minors XX: Wagner's conjecture, manuscript.

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N. Robertson and P. Seymour, Graph minors XX: Wagner's conjecture, manuscript.

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N. Robertson and P.D. Seymour. Graph minors XX: Wagner's conjecture. Journal of Combinatorial Theory, Series B, to appear.

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N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture.

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N. Robertson and P. Seymour, Graph minors XX: Wagner's conjecture, manuscript.

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N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture.

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N. Robertson and P.D. Seymour, Graph minors XX: Wagner's conjecture, manuscript.

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