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. 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 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.

Definitions

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

Subgraph Containment

Topological Order

Immersion Order

Minor Order

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

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.

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.

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

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.

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)

References Adler, Isolde, et al. "Faster parameterized algorithms for minor containment." Theoretical Computer Science (2011): Chen, Jianer, Iyad A. Kanj, and Ge Xia. "Improved parameterized upper bounds for vertex cover." Mathematical Foundations of Computer Science Springer Berlin Heidelberg, Chudnovsky, Maria, and Paul Seymour. "A well-quasi-order for tournaments." Journal of Combinatorial Theory, Series B (2011): Fellows, Michael R., and Michael A. Langston. "Nonconstructive tools for proving polynomial-time decidability." Journal of the ACM (JACM) 35.3 (1988): Kinnersley, Nancy G., and Michael A. Langston. "Obstruction set isolation for the gate matrix layout problem." Discrete Applied Mathematics 54.2 (1994): 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): Robertson, Neil, and Paul D. Seymour. "Graph minors. XX. Wagner's conjecture." Journal of Combinatorial Theory, Series B 92.2 (2004):

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.