The Graph Minor Theorem

CS594 GRAPH THEORY PRESENTATION SPRING 2014RON 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 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.

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

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 .

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.

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

𝐾 5

Under subgraph containment, if is isomorphic to a subgraph of

Subgraph Containment

𝐾 5

Subgraph Containment

𝐾 4≤𝐾 5

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 .

𝐶8 𝐶6 𝐶4

𝐶6≤𝑇 𝐶8 𝐶4≤𝑇𝐶6

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.

Immersion Order

𝐾 4≤𝑖𝑄3

Minor Order

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

Minor Order

𝑊 4≤𝑚𝑄3

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 ApplicationsVertex 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.

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

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

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)

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 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.

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.