Neil Robertson and P.D. Seymour; Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B 41(1986)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a minor of G j . Proof. If there is an i 2 such that G 1 is a minor of G i we are done. Otherwise fG i : i 2g is well quasi ordered by theorem 5.5 so there are 2 i j such that G i is a minor of G j . Theorem 5.5 easily follows from theorem 5. 4 and the following due to Robertson and Seymour [17]. Theorem 5.7. If H is a planar graph then the set of graphs with no H minor has bounded tree width. 8 Sketch proof. For n 1 let the n grid be a graph with vertex set f(i; j) 1 i; j ng and two vertices (i; j) and (i ; j ) are connected by an edge if and only if ji Gamma i j jj ....

....8 Sketch proof. For n 1 let the n grid be a graph with vertex set f(i; j) 1 i; j ng and two vertices (i; j) and (i ; j ) are connected by an edge if and only if ji Gamma i j jj Gamma j j = 1. It can be shown that the n grid has tree width n [16] The tree width theorem [17] shows that for each n there exists an m(n) 1 such that every graph with tree width m(n) has an n grid minor. Hence a graph has large tree width if and only if it has a large grid minor. Let H be a planar graph. There is a planar graph H with maximum degree 3 such that H is a minor of H ....

N. Robertson and P. D. Seymour. Graph minors V : excluding a planar graph. Journal of Combinatorial Theory B, 41:92--114, 1986.

....this with the results in [4] we obtain the following corollary. Corollary 1.2. For any planar graph H and nite eld F, the class of F representable matroids with no M(H) minor is well quasi ordered with respect to taking minors. For graphs such results were obtained by Robertson and Seymour [12]. Date: June 24, 2003. 1991 Mathematics Subject Classi cation. 05B35. Key words and phrases. branch width, matroids, graph minors, grids. This research was partially supported by grants from the Natural Sciences and Engineering Research Council of Canada and the Marsden Fund of New Zealand. ....

N. Robertson, and P. D. Seymour, Graph minors V: excluding a planar graph, J. Combin. Theory, Ser. B 41 (1986), 92-114.

....and Further Improvements As a consequence of Theorem 3.12, we establish an upper bound on the treewidth (or branchwidth) of a bounded genus graph that excludes some planar graph H as a minor. As part of their seminal Graph Minors series, Robertson and Seymour proved the following: Theorem 3. 13 ([33]) If G excludes a planar graph H as a minor, then the branchwidth of G is at most b H and the treewidth of G is at most t H , where b H and t H are constants depending only on H. The current best estimate of these constants is the exponential upper bound t H 2(2jV (H)j 4jE(H)j) 38] ....

N. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B, 41 (1986), pp. 92-114.

....and Further Improvements As a consequence of Theorem 3.12, we establish an upper bound on the treewidth (or branchwidth) of a bounded genus graph that excludes some planar graph H as a minor. As part of their seminal Graph Minors series, Robertson and Seymour proved the following: Theorem 3. 13 ([33]) If G excludes a planar graph H as a minor, then the branchwidth of G is at most b H and the treewidth of G is at most t H , where b H and t H are constants depending only on H. The current best estimate of these constants is the exponential upper bound t H 2(2 V (H) 4 E(H) 38] ....

N. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B, 41 (1986), pp. 92--114.

....graph G s;t obtained from G, s, and t by deleting every edge e not on a simple path from s to t. As we will see in more details in Section 3, the bound of Theorem 1. 1 derives from an optimal bound for square meshes, and from an upper bound of the excluding grid theorem of Robertson and Seymour [33]. This latter bound is likely far from best possible and, as mentioned in [19] Robertson, Seymour and Thomas [34] think that the upper bound might be exponentially improved so that the lower bound of Theorem 1.1 would be 309 ) Actually, for planar graphs, we derive a larger lower bound: ....

....of at least (k ) k) 144 k) bits. Since k = bminf 12 gc, log k = 398 minfp; qg) which completes the proof. 3 Proofs of Theorems 1.1 and 1.2 We use the excluding grid theorem of Robertson and Seymour, whose short proofs can be found in [18, 19] Theorem 3. 1 (Robertson Seymour [33]) For every integer r there is an integer k such that every graph of tree width at least k has an r r mesh as minor. 8 So, let us de ne f(r) as the smallest integer k satisfying Theorem 3.1. The constructive proof of the excluding grid theorem, given in [19] shows that f(r) 6 2 5r 5 log 2 ....

N. Robertson, and P.D. Seymour. Graph Minors. V. Excluding a planar graph. Journal of Combin. Theory B 41:92-114 (1986). 12

....any fixed graph H is a minor of an arbitrary n node graph [27] It follows from these two facts that there is an O(n ) time algorithm to test for membership in any minor closed set of graphs. Finally, when the minor closed family G does not contain all planar graphs, results of [26] and [25] (see [27] imply that one can test for membership in G in time O(n ) Thus, in our case, the minor closed family of graphs we are dealing with is the set of universally stable graphs, and by Theorem 2.10, this family does not contain all the planar graphs. Hence we have Theorem 3.17. There ....

N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. J. of Combinational Theory, Ser. B, 41:92--114, 1986.

....i , ffl for each v 2 V , the set of nodes fi 2 I j v 2 X i g induces a subtree of T. The width of a tree decomposition (fX i j i 2 Ig; T = I; F ) equals max (jX i j Gamma 1) The treewidth of a graph G is the minimum width over all tree decompositions of G. Robertson and Seymour proved in [14] (see also [15] that for any planar graph H there exist a constant c H such that any H 0 minor free graph has treewidth at most c H . Given a planar graph H, we define the minimum excluding bound of H, med(H) as the maximum treewidth over all H minor free graphs. as the minimum k bounding the ....

Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B, 41:92--114, 1986.

....k and l vertices. As usual for a functiuon we use j S to denote its restrictioin to the set S. 2. Parameterizing by treewidth Treewidth was first defined by Halin in [31] and was reintroduced independently in [3] and [46] It plays a key role in many proofs in structural graph theory [47, 42] and served as one of the cornerstone concepts for the lengthy proof of the Wagner s conjecture developed by Robertson and Seymour in their Graph Minor Series (see [45] for a survey) Formally, THE COMPLEXITY OF PARAMETERIZED H COLORINGS 5 Definition 2.1. A tree decomposition of a graph G is a ....

N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory (series B), 41:92--114, 1986.

....62. Proof of Theorem 62 Let C be a minor closed class of connected graphs that is not the class of all connected graphs and contains at least one graph with at least two cycles. We have two cases to consider: Case 1. The class C has bounded tree width. It follows from Robertson and Seymour [29] that some connected planar graph G is not in C. Let H 2 C with at least two cycles. By Lemma 63, there exists a covering K of H such that G K. We cannot have K 2 C, because we would have G 2 C, contradicting the choice of G. Hence C is not closed under connected coverings. Case 2. The class C ....

N. Robertson and P. Seymour. Graph minors v : excluding a planar graph. J. Combin. Theory, Ser. B, 35:92-114, 1986.

....it has high external connectivity. We observe that such graphs H must contain a large binary tree with some small additions, and prove that some canonical instances of this structure are also sucient to force a large complete minor. 1 Introduction A fundamental result of Robertson and Seymour [7] states that a graph has large tree width if and only if it contains a large grid minor. In their short proof of this theorem, Diestel et al. 3] introduced the concept of externally connected sets. A set X V (G) is externally k connected in G if jXj k and, for all subsets Y; Z X with jY j = ....

N. Robertson and P.D. Seymour, Graph minors V. Excluding a planar graph, J. Combin. Theory B 41 (1986), 92-114.

....given 2 colored grid can be mapped homomorphically to a given colored graph is NP complete and, if parameterized by the grid size, W[1] complete. We also prove a similar result for directed grids. These results are linked to tree width by the deep Excluded Grid Theorem due to Robertson and Seymour [12]. 2 . Preliminaries Relational Structures. A vocabulary is a finite set of relation symbols. In the following, always denotes a vocabulary. E always denotes a binary relation symbol and C 1 ; C 2 ; denote unary relation symbols. A structure A consists of a non empty set A, called the ....

....0 ; b 0 ) 2 E G . We call a minor map from H to G. We write H G to denote that H is a minor of G and : H G to denote that is a minor map from H to G. The connection between grids and tree width is made by the following deep Excluded Grid Theorem: Theorem 7 (Robertson and Seymour [12]) Let C be a class of graphs. Then C has bounded tree width if, and only if, there is a grid that is not a minor of any graph in C. 3 . Grid homomorphism For c 1, let c GRID be the class of all c colored square grids. We shall prove that c GRID HOMOMORPHISM is NP complete and that ....

N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92--114, 1986.

....Theorem 8 ( FG99a] Let C be a class of graphs such that there exists a graph that is not a minor of a graph in C. Then MC(C;FO) is in FPT. Robertson and Seymour proved that a class C of graphs has bounded treewidth if, and only if, there is a planar graph that is not a minor of a graph in C [RS86b]. Putting things together, we obtain a nice corollary: Corollary 2. Let C be a class of graphs that is closed under taking minors. 1) Assume that P 6= NP. Then MC(C;MSO) is in FPT if, and only if, C has bounded tree width. 2) Assume that FPT 6= W[1] Then MC(C;FO) is in FPT if, and only if, C ....

N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92-114, 1986.

....given 2 colored grid can be mapped homomorphically to a given colored graph is NP complete and, if parameterized by the grid size, W[1] complete. We also prove a similar result for directed grids. These results are linked to tree width by the deep Excluded Grid Theorem due to Robertson and Seymour [11]. 2 2 . Preliminaries Relational Structures. A vocabulary is a nite set of relation symbols. In the following, always denotes a vocabulary. E always denotes a binary relation symbol and C 1 ; C 2 ; denote unary relation symbols. A structure A consists of a non empty set A, called the ....

....0 ; b 0 ) 2 E G . We call a minor map from H to G. We write H G to denote that H is a minor of G and : H G to denote that is a minor map from H to G. The connection between grids and tree width is made by the following deep Excluded Grid Theorem: Theorem 7 (Robertson and Seymour [11]) Let C be a class of graphs. Then C has bounded tree width if, and only if, there is a grid that is not a minor of any graph in C. 3 . Grid homomorphism For c 1, let c GRID be the class of all c colored square grids. We shall prove that c GRID HOMOMOR PHISM is NP complete and that ....

N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92114, 1986.

....not capture polynomial time on all databases [6] it has turned out to be the right logic in several special cases. The best previously known result is that it captures polynomial time on the class of planar graphs [14] Our Theorem 1 is somewhat complementary by a result of Robertson and Seymour [22] saying that a class C of graphs is of bounded tree width if, and only if, there is a planar graph that is not a minor (see below) of any graph in C. We next turn to fixed point logic without counting. Although this weaker logic does certainly not capture polynomial time on databases of bounded ....

N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92--114, 1986.

....if it has small tree width. We further propose a very simple obstruction to small tree width inspired by that proof, showing that a graph has small tree width if and only if it contains no large highly connected set of vertices. 1 Introduction The following theorem of Robertson and Seymour [5] plays a fundamental role in their theory of graph minors: Theorem 1 Given any graph X, the graphs without an X minor have bounded tree width if and only if X is planar. Since planar grids can have arbitrarily large tree width (see below) the only if direction here is immediate: if X is ....

.... fat vertices and superimpose a drawing of a sufficiently fine grid) it suffices to show the following: Theorem 2 For every integer r there is an integer k such that every graph of tree width at least k has an r Theta r grid minor. Proofs of Theorem 2 have been given by Robertson Seymour [5], by Robertson, Seymour Thomas [7] and by Reed [3] All these proofs are long and technical. Our main purpose in this paper is to offer a short and selfcontained new proof of Theorem 2. This will be given in Section 3, which can be read independently of the rest of the paper. We remark that ....

N. Robertson & P.D. Seymour, Graph Minors. V. Excluding a planar graph, J. Combin. Theory B 41 (1986), 92--114.

....and the expression contains one factor a(p 1 ; pn ) for each hyperedge. We omit the details.2 The minimum cardinality of C C such that G = val(t) for some t 2 T (FHR (C) is connected with an important complexity measure on graphs and hypergraphs called treewidth. We refer the reader to [55] and to [20] Theorem 1.1 for more details. We only indicate that the minimum cardinality of C such that G = val(t) t 2 T (FHR (C) is MaxfCard( G) twd(G) 1g where twd(G) denotes the tree width of G. 47 A subset of HS(C) is called an HR set of hypergraphs i it is HS equational. The ....

....(or FHR system of equations) 6) a V R grammar (or F V R system of equations) 7) an MS 1 formula, 8) the set OBST (L) Proof : The set L is MS 1 de nable by Corollary 7. 3 (based on the Graph Minor Theorem) We have the following implications : 3) 4) by Robertson and Seymour [55] (3) 1) by Corollary 6.7 (2) since the set of graphs of tree width at most k is HR (see [20] 1) 2) by Theorem 5.15 (2) 1) by a result of [23] We now consider e ective constructions of such devices. 5) 6) because the proofs of Theorem 5.15 and [23] are e ective. 8) 7) ....

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: ROBERTSON N., SEYMOUR P., Graph minors V : Excluding a planar graph, J. 74 Comb. Theory Ser. B 52 (1986) 92-114.

....given 2 colored grid can be mapped homomorphically to a given colored graph is NP complete and, if parameterized by the grid size, W[1] complete. We also prove a similar result for directed grids. These results are linked to tree width by the deep Excluded Grid Theorem due to Robertson and Seymour [11]. 2 2 . Preliminaries Relational Structures. A vocabulary is a finite set of relation symbols. In the following, always denotes a vocabulary. E always denotes a binary relation symbol and C 1 ; C 2 ; denote unary relation symbols. A structure A consists of a non empty set A, called the ....

....that (a 0 ; b 0 ) 2 E G . We call a minor map from H to G. We write H G to denote that H is a minor of G and : H G to denote that is a minor map from H to G. The connection between grids and tree width is made by the following deep Excluded Grid Theorem: Theorem 7 (Robertson and Seymour [11]) Let C be a class of graphs. Then C has bounded tree width if, and only if, there is a grid that is not a minor of any graph in C. 3 . Grid homomorphism For c 1, let c GRID be the class of all c colored square grids. We shall prove that c GRID HOMOMOR PHISM is NP complete and that ....

N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92--114, 1986.

....that is hereditary with respect to min and such that there is a planar graph H with :E(H) there is an algorithm to test it in time O i jV j 2 j . The articles of the Graph Minors Series published in journals until now are Graph Minors I to X. These are [RS83a] RS86a] RS83b] RS90a] [RS86b], RS86c] RS88a] RS90c] RS90b] RS91] Until now XI to XVI circulate as manuscripts, these are [RS85b] RS86d] RS86e] RS87] RS88b] RS89] This theory can be used to solve several problems algorithmically. Besides giving a unified approach to many problems that have been solved ....

Neil Robertson and Paul Seymour, Graph minors V, excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92--114.

....algorithms, in particular for certain hard (NP complete) problems. They also arise in the study of the structure of graphs. For instance, treedecompositions have been introduced by Robertson and Seymour in their study of the structure of graphs that do not contain a xed graph as a minor ([GM1,GM5]) These decompositions yield the notion of tree width of a nite graph. The extension of the de nition to in nite graphs is straightforward. An in nite graph may have nite or in nite tree width. The compactness theorem of Thomassen [T] asserts that the tree width of an in nite graph is an ....

N. Robertson, P. Seymour, Graph Minors V, Excluding a planar graph, J. Comb. Th.B, 41 (1986) 92-114.

....tree width. Such a result was implicit already in the work of Baker [5] With a bound on tree width we can use dynamic programming techniques to compute many graph properties in linear time [8, 40] A result similar to the one in this section follows easily from the Robertson Seymour wall lemma [36] (Lemma 5 below) However we give the following direct proof to make explicit the dependence on the diameter, and to show that the result does not introduce any of the scary constants ubiquitous in Robertson Seymour theory. We first define the concept of tree width, introduced by Robertson and ....

.... In this section we exactly characterize those minor closed families of graphs having the diameter treewidth property, in a manner similar to Robertson and Seymour s characterization of the minorclosed families with bounded treewidth as being those families that do not include all planar graphs [36]. Definition 3. An apex graph [42] is a graph G such that for some vertex v (the apex) G v is planar. Apex graphs are also known as nearly planar graphs, and have been introduced to study linkless 3 dimensional embeddings of graphs [37] The significance of apex graphs for us is that they ....

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N. Robertson and P. D. Seymour. Graph minors V: excluding a planar graph. J. Combinatorial Theory B, 41:92--114, 1986.

....we show that universal stability is a decidable property; and in fact it can be decided in polynomial time. To prove this, we show that the set of universally stable graphs is closed under the taking of minors; polynomial time decidability then follows from results of Robertson and Seymour [22, 23]. 1.3 Preliminaries It is straightforward to formalize the model we have been discussing above. In every time step t, the current configuration C t of the system is a collection of sets fS t e : e 2 Gg, such that S t e is the set of packets waiting in the queue for e at the end of step t. ....

....any fixed graph H is a minor of an arbitrary n node graph [24] It follows from these two facts that there is an O(n 3 ) time algorithm to test for membership in any minor closed set of graphs. Finally, when the minor closed family G does not contain all planar graphs, results of [23] and [22] (see [24] imply that one can test for membership in G in time O(n 2 ) Thus, in our case, the minor closed family of graphs we are dealing with is the set of universally stable graphs, and by Theorem 2.10, this family does not contain all the planar graphs. Hence we have Theorem 3.17 There ....

N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. J. of Combinational Theory, Ser. B, 41:92--114, 1986.

....hence also a path from the left of W to the right. We omit the straightforward proof that this is a preference of order h 1. The reader should be able to convince herself that it is a preference of order h 1 3 . We actually need to apply a dual result of Robertson, Seymour, and Thomas[10] see [6] for a precursor) which is much more di cult: Theorem 2.2: For every even integer h 6, if G contains a preference B of order 20 64h 5 then G contains a wall W of height h. Furthermore, we can nd such a wall W which is controlled by B, i.e. such that for every subset X of at most h 1 ....

N. Robertson and P. D. Seymour, Graph Minors. V. Excluding a planar graph, Journal of Combinatorial Theory(B) 41 (1986), 92 - 114.

....minors. AMS classification: 68R10, 05C85, 05C05 1 Introduction 1. 1 Background The notions of tree decomposition and treewidth have received much attention recently, not in the least due to the important role they play in the deep results on graph minors by Robertson and Seymour (see e.g. [26, 27, 29, 30, 28], and many other papers in this series) See also [20] Also, many graph problems, including a very large number of well known NP hard problems, have been shown to be linear time solvable on graphs that are given together with a tree decomposition of treewidth at most k, for constant k. See, ....

N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B, 41:92--114, 1986.

....u, v, u Gamma v or v Gamma u getting a path P 0 of G violating the ranking condition, in contradiction to the choice of c. 2 Corollary 10 For each fixed t, the class of graphs satisfying r (G) t can be recognized in linear time. Proof: In [1] using results from Robertson and Seymour [22, 23], it is shown that every minor closed class of graphs that does not contain all planar graphs, has a linear time recognition algorithm. The result now follows directly from Lemma 9. 2 As regards edge rankings, a simple argument yields a much stronger assertion as follows. Theorem 11 For each ....

....This result implies polynomial time computability of the vertex ranking number for any class of graphs with a uniform upper bound on the treewidth, e.g. outerplanar graphs, series parallel graphs, Halin graphs. The notion of treewidth has been introduced by Robertson and Seymour (see e.g. [22]) Definition 14 A tree decomposition of a graph G = V; E) is a pair (fX i j i 2 Ig; T = I; F ) with X = fX i j i 2 Ig a collection of subsets of V , and T = I; F ) a tree, such that ffl S i2I X i = V ffl for all edges fv; wg 2 E there is an i 2 I with v; w 2 X i ffl for all i; j; k 2 ....

N. ROBERTSON, P. D. SEYMOUR. Graph minors. V. Excluding a planar graph. Journal on Combinatorial Theory Series B 41 (1986), 92--114.

....not capture polynomial time on all databases [6] it has turned out to be the right logic in several special cases. The best previously known result is that it captures polynomial time on the class of planar graphs [14] Our Theorem 1 is somewhat complementary by a result of Robertson and Seymour [22] saying that a class C of graphs is of bounded tree width if, and only if, there is a planar graph that is not a minor (see below) of any graph in C. We next turn to fixed point logic without counting. Although this weaker logic does certainly not capture polynomial time on databases of bounded ....

N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92--114, 1986.

....ffl for each v 2 V , the set of nodes fi 2 I j v 2 X i g induces a subtree of T. The width of a tree decomposition (fX i j i 2 Ig; T = I; F ) equals max i2I (jX i j Gamma 1) The treewidth of a graph G is the minimum width over all tree decompositions of G. Robertson and Seymour proved in [14] (see also [15] that for any planar graph H there exist a constant c H such that any H 0 minor free graph has treewidth at most c H . Given a planar graph H, we define the minimum excluding bound of H, med(H) as the maximum treewidth over all H minor free graphs. as the minimum k bounding ....

Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B, 41:92--114, 1986.

....that refines the well known modular decomposition. A graph complexity measure that we call clique width is associated in a natural way with this graph decomposition, much the same way tree width is associated with tree decompositions (which are actually hierarchical decompositions of graphs [14]) Hierarchical graph decompositions are interesting for algorithmic purposes. In the following, by a decomposition of a graph, we mean either a tree decomposition, or the unique modular decomposition, or a decomposition of the type that we shall define below. A decomposition of a graph G can be ....

....that if a set of graphs L has a decidable MS 1 theory (which is a weaker condition) then it is interpretable in a set of trees . This condition is equivalent by results in [10, 11, 13] to saying that L has bounded clique width. The result concerning MS 2 uses a result of Robertson and Seymour [14] asserting that graphs of large tree width contain large square grids as minors. At present, we lack a similar structural characterization of graphs with large clique width that would allow us to establish the conjecture. We hope that the investigation of clique width will yield such a result and ....

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N. Robertson and P. Seymour, Graph minors V, Excluding a planar graph, Journal of Combinatorial Theory (B), 41, (1986), 92--114.

....n grid has a minor isomorphic to H . B15. Theorem. For every planar graph H there exists an integer k such that if a graph G has tree width at least k, then it has an H minor. B16. Exercise. If H is non planar then no such integer exists. B17. Remark. The above theorem was first proved in [9]. A simpler proof with a better bound on k was given in [11] the bound there is k = 20 2n 5 , where n = 2jV (H)j 4jE(H)j. It seems to be an interesting and difficult problem to decide if k is bounded by some polynomial in n. B18. Exercise. Let H be the n Theta n grid, and let k be as in ....

N. Robertson and P. D. Seymour, Graph Minors V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92--114.

....As a very special case of Theorem 6.3, we get that the LOG PATH problem is in P. It is not difficult to check that all the algorithms we have described are easily parallelizable. It follows therefore that the LOG PATH problem is even in NC. As mentioned in the introduction, Robertson and Seymour [RS86b] showed that if C is a minor closed family of graphs that excludes at least one planar graph H , then there exists a (huge) constant c H such that every graph in C has tree width at most c H . As a simple corollary to Theorem 6.3, we get that if G = V; E) and H = V H ; EH ) such that jV H j = ....

N. Robertson and P. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92--114, 1986.

....we show that universal stability is a decidable property; and in fact it can be decided in polynomial time. To prove this, we show that the set of universally stable graphs is closed under the taking of minors; polynomial time decidability then follows from results of Robertson and Seymour [17, 18]. 1.2 Preliminaries It is straightforward to formalize the model we have been discussing above. In every time step t, the current configuration C t of the system is a collection of sets fS t e : e 2 Gg, such that S t e is the set of packets waiting in the queue for e at the end of step t. ....

....lengthy, and we omit it in this version. By results of Robertson and Seymour, this fact implies that the set of graphs that are not universally stable has only finitely many minor minimal elements. Moreover, the results of Section 2. 2 imply that not all planar graphs are universally stable; by [17, 18], we have the following. Theorem 3.9 There is an algorithm with running time O(n 2 ) that decides if a graph is universally stable. 4 Bounds on queue size for universally stable protocols The maximum queue size required by a queueing protocol is one of the main parameters determining its ....

N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. J. of Combinational Theory, Ser. B, 41:92--114, 1986.

....the problem of identifying a very broad series of classes of graphs, namely those closed under taking graph minors. Such sets of graphs have been studied for a number of years by graph theorists, most notably in a series of papers by Robertson and Seymour (see for example among other papers [12,14,13,16,15]) The classes of graphs that are closed under taking minors are very common. Examples are the planar graphs, graphs which can be embedded in 3 dimensional space without knots, graphs which can be embedded in 3 dimensional space without interlocking cycles, graphs with a fixed This work has ....

N. Robertson and P.D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B, 41 (1986) 92--114.

....for the class of graphs of tree width at most k for every fixed k. Given graphs G and H, G is a minor of H, denoted G H, if G can be obtained from a subgraph of H by contracting edges. If a G is not a minor of H, then H is a graph with no G minor. An important result of Robertson and Seymour [5] (see also [8] is that if P is a planar graph, then there is an integer k such that every graph G with no P minor has tree width at most k. This is not true for any non planar graph, as the n n planar grid has tree width n (see [4] The authors [2] generalizing a relaxation of a conjecture ....

....that if K is a planar graph, then #(K) #, and a graph with no K minor may be constructed by clique joins, starting from graphs on at most w(K) vertices. This special case of Theorem 2.1 (or Corollary 2. 2) that, for every planar graph P , a graph with no P minor has low tree width, appears in [5] (see also [8] 3. Layers The main step towards proving Theorem 1.2 is finding an edge and a vertex (j 1) coloring, of a (#, r) outgrowth (G, H) such that any j colors form a graph with bounded tree width. The bound would involve #, r and j. Clearly there will be no loss of generality in ....

N. Robertson and P.D. Seymour, Graph Minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986) 92--114.

....that none of the graphs G 2 , G 3 , contains G 1 as a minor. Robertson and Seymour then prove that these graphs have a special structure. In particular, if G 1 is a forest, then the graphs have bounded path width [30] If G 1 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 additional definition is needed. Let G be a graph, S a surface, and k an integer. We say that G can be ....

....result. Theorem 4.3 (Thomassen [60] Let G # Forb(S g ) 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, 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 be positive integers, and ....

[Article contains additional citation context not shown here]

N. Robertson, P. D. Seymour, Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986) 92--114.

....and the tree width of G is the minimum width of a tree decomposition of G. Tree width was introduced in [5] and, independently, in [2] under the name partial k tree. It has since received widespread attention, for the following reasons: i) It serves as a cornerstone of the Graph Minors theory [5, 6], ii) it can be used to prove theorems in structural graph theory [4, 6] iii) it has many algorithmic applications due to the fact that many NP hard problems can be solved in linear time when restricted to graphs of bounded tree width [1, 2, 7] iv) it has been successfully used in practical ....

....G. Tree width was introduced in [5] and, independently, in [2] under the name partial k tree. It has since received widespread attention, for the following reasons: i) It serves as a cornerstone of the Graph Minors theory [5, 6] ii) it can be used to prove theorems in structural graph theory [4, 6], iii) it has many algorithmic applications due to the fact that many NP hard problems can be solved in linear time when restricted to graphs of bounded tree width [1, 2, 7] iv) it has been successfully used in practical computations [3] In Section 2 of this paper we generalize tree width to ....

[Article contains additional citation context not shown here]

N. Robertson and P. D. Seymour, Graph Minors V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92--114.

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Neil Robertson and P.D. Seymour; Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B 41(1986)

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N. ROBERTSON, P. D. SEYMOUR. Graph minors. V. Excluding a planar graph. Journal on Combinatorial Theory Series B 41 (1986), 92--114.

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N. Robertson, P.D. Seymour, Graph minors V: Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92-114.

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N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. J. of Combinational Theory, Ser. B, 41:92--114, 1986.

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N. Robertson and P.D. Seymour. Graph minors V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41:92--11, 1986.

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N. Robertson and P. D. Seymour: Graph minors V. Excluding a planar graph. J. Comb. Theory Series B 41, pp 92-114, 1986.

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N. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, Journal of Combinatorial Theory Series B, 41 (1986), pp. 92--114. 11

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N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory (series B), 41:92--114, 1986.

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N. ROBERTSON AND P. D. SEYMOUR, Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B, 41 (1986), pp. 92--114.

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N. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, J. Comb. Theory Series B, 41 (1986), pp. 92-114. 13

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N. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, J. Comb. Theory Series B, 41 (1986), pp. 92--114.

No context found.

N. Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory (series B), 41:92-114, 1986.

No context found.

N. Robertson, and P.D. Seymour. Graph Minors. V. Excluding a planar graph. Journal of Combin. Theory B 41:92-114 (1986).

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N. Robertson & P.D. Seymour, Graph minors. V. Excluding a planar graph, J. Combin. Theory B 41 (1986), 92--114. 26

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N. Robertson, and P.D. Seymour. Graph Minors. V. Excluding a planar graph. Journal of Combin. Theory B 41:92-114 (1986).

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