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The Graph Minor Theorem

The Graph Minor Theorem. CS594 Graph theory presentation Spring 2014 Ron Hagan. Introduction. Neil Robertson, Paul Seymour published a series of papers in the Journal of Combinatorial Theory Series B.

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The Graph Minor Theorem

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  1. The Graph Minor Theorem CS594 Graph theory presentation Spring 2014 Ron Hagan

  2. Introduction • Neil Robertson, Paul Seymour published a series of papers in the Journal of Combinatorial Theory Series B. • Beginning with Graph Minors.I.Excluding a Forest, appearing and 1983 and currently up to Graph Minors.XXIII.Nash-Williams’ Immersion Conjecture. The most recent appearing in 2012. • One of the main intended results culminated in Graph Minors.XX.Wagner’s Conjecture, in a proof of what is now known as The Graph Minor Theorem.

  3. Definitions • A binary relation on a set is a quasi-order if it is both reflexive and transitive. • For all , • (reflexive) • If and , then (transitive) • A partial-order is a quasi-order that also requires anti-symmetry, that is: • If and , then

  4. A set is well-quasi-ordered (wqo) under a relation if: • 1) It is well-founded. Every non-empty subset has a minimal element. • 2) It does not contain any infinite antichains. For all infinite sequences of elements from there is such that .

  5. Well-quasi-orders have also been described in terms of ideals (see for example Higman or J. Kruskal). • A subset of is called an upper ideal if and implies . • If , then is said to generate or is the ideal generated by . • In this context, a space is well-quasi-ordered if it is quasi-ordered and every ideal has a finite generating set.

  6. Orders on Sets of Graphs • Some potential orders on the set of finite undirected graphs: • Subgraph Containment • Topological Order • Immersion Order • Minor Order

  7. Subgraph Containment Under subgraph containment, if is isomorphic to a subgraph of

  8. Subgraph Containment

  9. Subgraph Containment

  10. Topological Order • A graph is a subdivision of a graph if can be obtained by subdividing edges of . In the topological order, if contains a subgraph isomorphic to a subdivision of .

  11. Immersion Order • In the immersion order, if there is a map and a map that takes each edge of to a path from and in such that paths given by are edge disjoint. • Equivalently, H is isomorphic to a subgraph obtainable from by a series of liftings.

  12. Immersion Order

  13. Minor Order • Allowable operations are taking subgraphs and contracting edges. if is (isomorphic to) a minor of .

  14. Minor Order

  15. The Graph Minor Theorem • The class of all finite undirected graphs is a wqo under the minor relation.

  16. Consequences and Applications • If a family of graphs is closed under taking minors, then membership in that family can be characterized by a finite list of minor obstructions.

  17. Consequences and Applications Vertex Disjoint Paths: Given a graph and a set of pairs of vertices of , does there exist paths in , mutually vertex-disjoint, such that joins and for ? If k is in the input of the problem, it is NP-complete. (Karp) In Graph Minors.XIII.The Disjoint Paths Problem, Robertson and Seymour give a algorithm for fixed k. As a consequence, they obtain a algorithm for checking minor containment.

  18. Consequences and Applications • If a family of graphs is closed under taking minors, then membership in that family can be tested in polynomial time. • Problems: • 1) The algorithm is non-constructive. (requires knowledge of obstruction set) • 2) It hides huuuuuuuuge constants of proportionality.

  19. Consequences and Applications • Dr. Langston and Mike Fellows pioneering work in applications included proofs that: • For every fixed k, gate matrix layout is solvable in polynomial time. • As well as analogs for: • Disk dimension • Minimum cut linear arrangement • Topological bandwidth • Crossing number • Maximum leaf spanning tree • Search number • Two dimensional grid load factor

  20. Consequences and Applications • Their work would lay the foundation for what would be formalized as a new field of study – fixed parameter tractability. • R.G. Downey and M.R. Fellows. Parameterized Complexity. Springer-Verlag 1999.

  21. Current Research • Improving minor containment checking. • Currently for branchwidth k: algorithm by Adler, Dorn, Fomin, San, and Thilikos.

  22. Current Research • Improving the cost of the hidden constant. • Best vertex cover time is due to Chen, Kanj, and Xia.

  23. Current Research • Identification of obstruction sets. • Obstruction set for 2-track GML consists of and • Obstruction set for 3-track GML contains 110 elements. (Kinnersley and Langston)

  24. Current Research • Extension of results to directed graphs. • Difficult to determine what a minor of a directed graph should be. • Work has been done on immersions of directed graphs. • The class of directed graphs is not a wqo under (weak) immersion. • BUT • The class of all tournaments is a wqo under strong immersion. (Chudnovsky and Seymour)

  25. References • Adler, Isolde, et al. "Faster parameterized algorithms for minor containment." Theoretical Computer Science 412.50 (2011): 7018-7028. • Chen, Jianer, Iyad A. Kanj, and Ge Xia. "Improved parameterized upper bounds for vertex cover." Mathematical Foundations of Computer Science 2006. Springer Berlin Heidelberg, 2006. 238-249. • Chudnovsky, Maria, and Paul Seymour. "A well-quasi-order for tournaments." Journal of Combinatorial Theory, Series B 101.1 (2011): 47-53. • Fellows, Michael R., and Michael A. Langston. "Nonconstructive tools for proving polynomial-time decidability." Journal of the ACM (JACM) 35.3 (1988): 727-739. • Kinnersley, Nancy G., and Michael A. Langston. "Obstruction set isolation for the gate matrix layout problem." Discrete Applied Mathematics 54.2 (1994): 169-213. • Langston, Michael A. “Fixed-Parameter Tractability, A Prehistory,” in The Multivariate Complexity Revolution and Beyond: Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday (H. L. Bodlaender, R. Downey, F. V. Fomin and D. Marx, editors), Springer, 2012, 3–16. • Robertson, Neil, and Paul D. Seymour. "Graph minors. XIII. The disjoint paths problem." Journal of Combinatorial Theory, Series B 63.1 (1995): 65-110. • Robertson, Neil, and Paul D. Seymour. "Graph minors. XX. Wagner's conjecture." Journal of Combinatorial Theory, Series B 92.2 (2004): 325-357.

  26. Homework • 1. Show that finite nondirected graphs are not wqo under subgraph containment. • 2. Show that finite nondirected graphs are not wqo under the topological order.

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