Graph Minors

Austin WyerMarch 23, 2016

Definitions

Definition of Graph Minor

Definition: We consider a graph H to be a minor of a graph G if we can obtain H from G by:

-Deleting edges

-Deleting vertices

-Contracting edges

Edge Contraction

Definition: A contraction of an edge e with endpoints u and v replaces these vertices with a new vertex w, such that w is adjacent to all vertices adjacent to both u and v.

By Claudio Rocchini - Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=1989074

Example - K3,3 as minor of Petersen Graph

By Pablo Angulo using Sage software - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=12026419

History

Klaus Wagner

● German

● March 31, 1910 - February 6,

2000

● Wagner’s Theorem 1937

● Wagner’s Conjecture

From: Professor Ulrich Faigle [30]

Wagner’s Theorem

Theorem: A graph is planar if and only if it does not contain K

3,3 or K

5 as a minor.

● Similar result to Kuratowski’s Theorem - Forbidden graph characterizations

● Difference in definition of containment - Minor vs Topological minor

Topological Minor

Definition: A graph H is a topological minor of a graph G if a subdivision of H is isomorphic to a subgraph of G.

● Every topological minor of a graph is also a minor. ● Converse is not always true. ● Does hold if the minor has a maximum degree of less than

or equal to 3.

Wagner’s Conjecture

Conjecture: Every infinite sequence of graphs G1

,G2

,... contains the distinct graphs G

i and G

j such that i < j and G

i is a

minor of Gj.

● Another way to state is that graphs are well-quasi-ordered when partially ordered under the graph minor relation.

● Eventually proved by Neil Robertson and Paul Seymour (We’ll talk about this later)

Well-quasi-ordering

Definition: A well-quasi-ordering is a binary relation that is both reflexive and transitive (Quasi-ordered). Additionally, any infinite sequence of elements contains a pair g

i and g

j such

that gi ≤ g

j where i < j (No infinite antichains).

Partial Order: a quasi-ordering with antisymmetry.

State of Graph Minors

Neil Robertson and Paul Seymour

● Professors at The Ohio State University and Princeton respectively.

● Provided alternative proof of the Four-Colour Theorem

● Together proved the Graph Minor Theorem

From Professor Richard Stanley [31]

Graph Minor Theorem

Theorem: Graphs are well-quasi-ordered when partially ordered by the graph minor relation.

Another important statement of this theorem is as follows

Theorem: For any family of graphs that is closed under the minor operation there is a finite set of forbidden minors.

Forbidden Minors

A family of graphs is said to be minor-closed if for any graph G in the family, any minor of G is also in the family.

The graph minor theorem tells us that there is a finite set of minors that characterize the minimal graphs not in the family.

We call this finite set of graphs the obstruction set.

Obstruction Sets

We already know one obstruction set for the planar graphs.

Robertson and Seymour showed you can find obstruction sets for any surface.

Unfortunately proof is nonconstructive

Important algorithmic implications

Graph Minor Theorem - Tools

Graph minor theorem important for more than just its results. Proof presented by Robertson and Seymour includes important tools in defining graph structure.

We’ll go on to describe some of these tools

Tree Decomposition

Definition: A tree decomposition of a graph G is made up of a tree T, and sets V

t that are associated with each node t in T.

The tree and its sets must satisfy these properties:

● Every vertex in G belongs to a set Vt

● For every edge in G there is a set Vt that contains both ends of the edge

● Consider the nodes t1

, t2

, and t3

. If t2

exists on a path from t

1 to t

3, and a vertex v is in the set V

t1 and V

t3, then it must

also exist in Vt2

Example Tree Decomposition

From Bockmayr and Reinert[21]

Tree-width

Definition: The width of a tree decomposition is the size of the largest set of the tree decomposition minus one.

Definition: The tree-width of a graph G is the minimum width of all possible tree decompositions of G.

Path Width

We say that a tree decomposition is a path decomposition if the tree T is a path.

We then call the width metric on such a decomposition the path-width.

Tree-width and Minors

It can be shown that contracting and deleting cannot increase treewidth, thus if H is a minor of G, then H’s tree-width can be at most the tree-width of G.

This gives rise to an important result

Theorem: For every positive integer k, the graphs with treewidth less than k are well-quasi-ordered under the minor relation.

Brambles

Bounded tree-width gives us an ordering on some graphs, but what about other graphs?

Robertson and Seymour introduced Brambles for such cases

Definition: A bramble B of a graph G is a set of subgraphs, such that each subgraph in the set either shares a vertex, or one of the vertices from each set are adjacent by an edge in G.

Example of Bramble on 3x3 Grid

By David Eppstein - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=20487288

Brambles and Tree-width

Definition: The order of a bramble is defined as the minimum number of vertices from the graph G that is needed to cover that bramble. A set of vertices covers a bramble if the intersection between that set and every set in the bramble is nonempty.

Theorem: Let k ≥ 0 be an integer. A graph has tree-width ≥ k if and only if it contains a bramble of order > k.

What is the order of the Bramble?

By David Eppstein - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=20487288

Brambles, Tree-width, and Planarity

From the previous example and theorem we know the 3x3 grid has a tree-width of at least 3. In fact, it’s tree-width is 3.

Theorem: Any nxn grid has a tree-width of size n.

From this theorem we can see that the planarity of a graph does not place a bound on the tree-width of the graph.

Brambles, Tree-width, and Planarity

Theorem: The following three statements are equivalent

1. F is a minor-closed family of bounded tree-width graphs2. One of the forbidden minors from the finite obstruction

set that characterizes F is planar. 3. F is a minor-closed family that does not include all planar

graphs.

Grids and Bounds on Tree-width

Theorem: For every integer n there is an integer k such that every graph of tree-width at least k has nxn grid minor.

Graphs have bounded tree-width if and only if a forbidden minor is planar.

We can use grids to help set these bounds, as any planar graph can be found to be the minor of some grid.

Forbidden Minors for fixed Tree-width

k = 1

k = 2

k = 3By David Eppstein - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=3016440

Finding Graph Minors

Theorem: For any fixed graph H, there exists a polynomial time algorithm for determining if a graph G contains H as a minor.

Original algorithm by Robertson and Seymour is O(n3)

Improvements have since been made to improve to O(n2)

What does that mean?

Because we are guaranteed by the graph minor theorem that the size of the obstruction set will be finite, then we have a polynomial time algorithm to test for membership in any minor-closed family.

We know the size of the obstruction set is finite, so we simply check for the existence of every forbidden minor.

Limitations

Though we are guaranteed the existence of such an algorithm, we must remember the proof is nonconstructive.

Thus, we need to know the obstruction set before we can test for it.

Example: Apex Graphs

Apex Graphs

Definition: Graphs that can be made planar by the removal of a single vertex. An apex is a vertex in the graph that can be removed to make the graph planar. All planar graphs are considered apex graphs.

Apex graphs are closed under the minor operation, and thus have a set of forbidden minors.

The complete set is currently unknown.

Other Limitations

We must also consider the size of the obstruction set. Though we know its size is fixed, it could be very large.

Consider the forbidden minors for a bounded tree-width k. The lower bound for the size of this obstruction set is an exponential of the square-root of k.

For large tree-width bounds this presents two major problems: computational cost and finding the forbidden minors.

Vertex Disjoint Paths Problem

Definition: Given a graph G and k pairs of vertices (s1

,t1

) … (sk,

tk), are there k mutually vertex disjoint paths such that path i

links the pair (si,t

i)?

This problem is known to be NP-Complete if k is an input to the problem.

Adding a Fixed Parameter

Theorem: For any fixed integer k, there is a polynomial time algorithm to decide the k-disjoint path problem.

Reduce problem of finding disjoint paths to whether a minor H exists in the graph.

Robertson and Seymour presented an O(n3) algorithm

Later improved by Kawarabayashi to O(n2)

Future Work

Current Research

● Simplifying proof of Robertson-Seymour theorem● Constructive Proofs - Find obstruction sets● Applying graph minor theorem to directed graphs

Hadwiger’s Conjecture

Conjecture: If all colorings of a graph G use k or more colors, then G contains the minor K

k.

Assumes that you can find k disjoint connected subgraphs, such that each subgraph is connected to every other subgraph. We then contact the subgraphs to obtain K

k.

Robertson, Seymour and Thomas proved for k = 6. Unknown for larger values.

References[1] https://en.wikipedia.org/wiki/Graph_minor

[2] https://en.wikipedia.org/wiki/Robertson–Seymour_theorem

[3] https://en.wikipedia.org/wiki/Edge_contraction

[4] https://en.wikipedia.org/wiki/Klaus_Wagner

[5] https://en.wikipedia.org/wiki/Wagner%27s_theorem

[6] https://en.wikipedia.org/wiki/Well-quasi-ordering

References[7] https://en.wikipedia.org/wiki/Partially_ordered_set

[8] https://en.wikipedia.org/wiki/Neil_Robertson_(mathematician)

[9] https://en.wikipedia.org/wiki/Paul_Seymour_%28mathematician%29

[10] https://en.wikipedia.org/wiki/Forbidden_graph_characterization

[11] https://en.wikipedia.org/wiki/Tree_decomposition

[12] https://en.wikipedia.org/wiki/Treewidth

References[13] https://en.wikipedia.org/wiki/Pathwidth

[14] https://en.wikipedia.org/wiki/Bramble_(graph_theory)

[15] https://en.wikipedia.org/wiki/Apex_graph

[16] https://en.wikipedia.org/wiki/Hadwiger_conjecture_(graph_theory)

[17] https://en.wikipedia.org/wiki/Parameterized_complexity#FPT

[18] Reinhard Diestel, “Graph Theory Electronic Edition 2000”(2000)

References[19] Ken-ichi Kawarabayashi and Bojan Mohar, “Some Recent Progress and Applications in Graph Minor Theory” (2006)

[20] Siddharthan Ramachandramurthi, “The Structure and Number of Obstructions to Treewidth”(1997)

[21] A. Bockmayr and K. Reinert, “Discrete Math for Bioinformatics WS: Tree Decompositions”(2010)

[22] Neil Robertson and Paul Seymour “Graph Minors. III. Planar tree-width”(1984)

References[23] Neil Robertson and Paul Seymour, “Graph Minors. V. Excluding a planar graph”(1986)

[24] Neil Robertson and Paul Seymour, “Graph Minors. IV. Tree-width and well-quasi-ordering(1990)

[25] Neil Robertson and Paul Seymour, “Graph Minors. VIII. A Kuratowski theorem for general surfaces”(1990)

[26] Neil Robertson and Paul Seymour, “Graph Minors. X. Obstructions to tree-decomposition”(1991)

References

[27] Neil Robertson and Paul Seymour, ”Graph Minors. XIII. The disjoint paths problem”(1995)

[28] Neil Robertson and Paul Seymour, ”Graph Minors. XVI. Excluding a non-planar graph”(2003)

[29] Neil Robertson and Paul Seymour, ”Graph Minors. XX. Wagner’s conjecture”(2004)

[30] http://www.ams.org/samplings/feature-column/fcarc-coloring2

[31] http://www-math.mit.edu/~rstan//photos/index.html

Homework

1. Provide a tree decomposition for the Petersen graph. 2. Prove that graphs are partially ordered by the minor

relationship.3. Show that an nxn grid has a tree-width of at least n.

(Hint- Consider using brambles)