Robertson-Seymour Theorem
A
project
submitted
by
Aabhas Majumdar
Roll no. -201202038
Content
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Introduction
Graph: What is it?
Definitions
Graph Minor
Kuratowski's Theorem
Wagner’s Conjecture
Well-Quasi-Order
Tree Decompostion
Tree width
Pathwidth
Robertson-Seymour theorem
Or
Graph Minor Theorem
Minor Closed Families
Disjoint Path
Limitation
Challenges to Practicality
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Conclusion
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Introduction
The deepest and likely the most important work in graph
theory is the Graph Minor Theory developed by Robertson and
Seymour in a series of over 20 papers spanning over 20 years.
They have led the way in developing a vast body of work in
graph theory. The celebrated result is a proof of an old and
intractable conjecture in graph theory, previously known as
Wagner's Conjecture, and now known as the Graph Minor
Theorem. The purpose of this project is to give the basic
concepts required to understand the theorem and few of its
main application.This project will not include its proof as it
requires a lot of tough mathematical tools and will take
hundreds of pages.
Signicantly, many of the tools used in the proof of the Graph
Minor Theorem can be applied to a very broad class of
algorithmic problems. For example, Robertson and Seymour
have obtained a relatively simple polynomial-time algorithm
for the disjoint paths problem ,a task that had eluded
researchers for many years. Other applications include
combinatorial problems from several domains, including
network routing,utilization and design. Indeed, it is a critical
measure of the value of the Graph Minor Theorem that so
many applications are already known for it. Only the tip of the
iceberg seems to have surfaced thus far.
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Graph
A finite simple graph G is defined by two finite sets V (G) and E(G),
where an element of V (G) is called a vertex and an element of E(G) is
an edge. Each edge is a two element subset of vertices. An edge e from
vertex u to vertex v is usually written as uv, and we say that e is
incident with u and v.
A multigraph is similar to a simple graph, but loops uu and multiple
edges from u to v are allowed.
A vertex degree of v denoted by d(v) is the number of edges of G
incident with v, with loops counting twice. Intuitively, this is a count of
how many ends of edges meet at v.
Next are a few basic definitions needed to know graph minor theorem.
Definitions
1)
Contraction of an edge- In graph theory, an edge contraction is an
operation which removes an edge from a graph while
simultaneously merging the two vertices it previously connected.
Edge contraction is a fundamental operation in the theory of graph
minors.
or
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Let G=(V,E) be a graph (or directed graph) containing an edge
e=(u,v) with u≠v.
Let f be a function which maps every vertex in
V\{u,v} to itself, and otherwise, maps it to a new vertex w. The
contraction of e results in a new graph G′=(V′,E′), where
V′=(V\{u,v})∪{w}, E′=E\{e}, and for every x∈V, x′=f(x)∈V′ is incident
to an edge e′∈E′ if and only if, the corresponding edge, e∈E is
incident to x in G.
2)
3)
4)
Sub graph- A sub graph of a graph G is a graph whose vertex set is
a subset of that of G, and whose adjacency relation is a subset of that
of G restricted to this subset.
Graph embedding- An embedding of a graph G on a surface Σ is a
representation of G on Σ in which points of Σ are associated to
vertices and simple arcs are associated to edges in such a way that:
 The endpoints of the arc associated to an edge e are the points
associated to the end vertices of e.

No arcs include points associated with other vertices.
 Two arcs never intersect at a point which is interior to either of
the arcs.
Complete graph- A complete graph is a simple undirected graph in
which every pair of distinct vertices is connected by a unique edge.
The complete graph on n vertices is denoted by Kn
Eg. K5 –
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5)
Bipartite Graphs- Bipartite graphs may be characterized in several
different ways:
 A graph is bipartite if and only if it does not contain an odd cycle.
 A graph is bipartite if and only if it is 2-colorable, (i.e. its
chromatic number is less than or equal to 2).
 The spectrum of a graph is symmetric if and only if it's a bipartite
graph
6)
Complete Bipartite Graphs- A complete bipartite graph or
biclique is a special kind of bipartite graph where every vertex of
the first set is connected to every vertex of the second set.
The complete bipartite graph with partitions of size |V1|=m and
|V2|=n, is denoted Km,n.
Eg.- K3,3
7)
Tree- A tree is a connected graph with no cycles(acyclic).
A tree G has following property : G is connected and has no cycles.
 G has no cycles, and a simple cycle is formed if any edge is added
to G.
 G is connected, but is not connected if any single edge is removed
from G.
 Any two vertices in G can be connected by a unique simple path.
 If G has n vertices it has n-1 edges.
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Two of the key notions required for Robertson Seymour theorem are
minors, and well-quasi-orders, and we examine them in that order.
Graph Minor
Given graphs H and G, we say that H is a minor of G (or that G contains
H as a minor) if a graph isomorphic to H can be obtained by removing
from G some vertices and edges and then contracting some edges in the
resulting subgraph. Thus every graph is a minor of itself, and the single
vertex graph is a minor of every non-empty graph.
For a slightly less trivial example, see the above figure, which
illustrates that the wheel with four spokes (W4) is a minor of the
binary three-cube (Q3).
Note: A minor is independent of the order of operations performed and
therefore can simply be characterized by a list of edge contractions and
deletions.
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Before going to well-quasi-order let’s study a related theorem called
Kuratowski's theorem and a conjecture called Wagner’s conjecture.
We know that K5 and K3,3 cannot be embedded on a 2-D plane. So are
there any more graphs of this type? This theorem and conjecture gives
a family of them.
Kuratowski's Theorem
Kuratowski’s Theorem states that “A graph is planar if and only if it
does
not contain a sub graph to K5 or K3;3. ”
Kuratowski’s theorem was refined by Wagner in his 1935 PhD Thesis
as Wagner conjecture.
Wagner conjecture
Wagner conjecture states that “A graph G is planar if and only if K5
and K3, 3 are not minors of G.”
Wagner’s work led to a more general study of families of graphs with
forbidden minors. A graph H is a forbidden minor for a set F of graphs if
H is not a minor of any graph in F. A forbidden minor H of F is minimal
if no proper minor of H is also a forbidden minor.
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Well-quasi-order
A class Q, equipped with a transitive and reflexive relation ≤ is called a
quasi-order. There has been some confusion as to the difference
between quasi-orders and partial-orders; it suffices to look at graph
minors to understand this difference . It is convenient to regard
isomorphic copies of a given graph as different entities and so, for
distinct graphs G and H, we can simultaneously have G ≤ H and H ≤G.
Thus ≤ is not a partial-order, because the minor relation is not antisymmetric.
A quasi-order with class Q and relation ≤ is a well-quasi-order if
(1) For every infinite sequence a1; a2;........... of elements of Q, there
exist integers 1 ≤ i < j such that ai ≤ aj , and
(2)
There exists no infinite descending chain b1 > b2 > ............. of
distinct elements of Q.
Next we discuss the important graph parameters treewidth and
pathwidth.
Tree Decomposition
The tree decomposition of a graph G is a tree T satisfying the following
properties:
• V (T) ⊆ P(VG), i.e., the nodes of T are subsets of VG.
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• For each edge (u, v) ∈ EG there is a node Xi of T, such that both u, v ∈
Xi.
• For each vertex u ∈ VG, the sub graph of T induced by sets containing
u is nonempty and connected (i.e. it is a sub tree).
Treewidth
Treewidth plays a critical role in the graph minors’ project. It may be
said that it measures the complexity of a graph, in the sense that a
graph of small treewidth can be recursively decomposed, by removing
a few vertices, into two graphs of roughly equal size. A consequence of
this is that many NP-hard problems can be efficiently solved in graphs
of small treewidth with dynamic programming.
The treewidth of a graph G, denoted by tw(G), is the minimum width
among all its decompositions.
Pathwidth
In the development of the graph minors project, treewidth was
preceded by another graph parameter, pathwidth. The pathwidth of a
graph can be much larger than its treewidth.
Several important applications of pathwidth arose well before the
graph minors project. The definition of pathwidth is similar to that of
treewidth, except that we restrict ourselves to tree decompositions
(T,X) where T is a path (such tree decompositions are called path
decompositions). Thus if (T,X) is a path decomposition of G, then every
vertex v of G is mapped into a subpath Pv of T (i.e., each vertex
essentially is mapped into an interval), so that whenever {u, v} is an
edge of G, then Pu and Pv intersect. The width of the path
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decomposition is the maximum number of subpaths Pv that are
incident with any vertex of T minus one.
In terms of graph minors, the most important theorem involving
pathwidth is
“For every forest F there is a number p(F), such that if a graph G does
not have a minor isomorphic to F, then G has pathwidth less than p(F).”
The Robertson-Seymour Theorem
Or
Graph Minor Theorem
We can now state the Graph Minor Theorem and some of the most
important consequences arising from its proof. Very little progress had
been made on this result, formerly a conjecture attributed to K.
Wagner, until the work of Robertson and Seymour.
The Graph Minor Theorem- The class of all graphs is a well-quasiorder under the minor relation.
There are many ways of stating Graph minor theorem or Robertson
Seymour theorem one of them similar to Wagner’s conjecture and one f
them related to order theory. Here is another way of stating it:“For every infinite sequence G1, G2 , . ..... of graphs, there exist distinct
integers i < j such that Gi is a minor of Gj .”
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While proving this theorem Robertson and Seymour also proved few
more theorems that were required to prove Graph Minor Theorem.
Here are they:1) Perhaps this Theorem is the most important work in Graph
minor areas:“For every fixed graph H, there exists an O(n3) algorithm for
deciding if a given graph of order n contains H as a minor.”
The constant hidden in the big O is a very rapidly growing
function of the size of H.
2) A graph G is planar if and only if G does not contain K5 or K3,3 as a
minor.
3) For every minor-closed family of graphs, the set of forbidden
minors is finite.
4) Let S be a given surface. Then graph embedability in S can be
tested in polynomial time. This theorem is a result of 1)
5) The disjoint paths problem can be solved in polynomial time for
every
fixed k.
The disjoint paths problem is addressed in more detail later.
6) For each positive integer k, there is an integer f(k) such that
every graph with tree-width of at least f(k) has a k-grid minor.
Minor-Closed Families
Robertson and Seymour also defined a new family of graph called
Minor closed family. Here is the definition :
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“It is a family of graph which doesn’t contains a specific set of graphs as
minors.”
The set of minors which is not contain in family is called forbidden
set.
Graph minor theory also states a Theorem of Minor closed families :“A family of graphs is minor-closed if and only if it has a finite number
of minimal
forbidden minors.”
The set of minimal forbidden minors is sometimes called an
obstruction set for the graph family.
We already have seen a minor closed family of planar graph in
Wagner’s theorem
With obstruction set been K5 & K3,3. Here a table of minor closed family
with their obstruction set :-
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Disjoint Paths
We are given, in a graph G, vertices si and ti (1 ≤ i ≤ k), not necessarily
distinct. We seek pair wise vertex-disjoint paths between si and ti (1 ≤ i
≤ k). In this section we outline how graph minors theory yields an
algorithm with complexity O(n3) for this problem, for each fixed value
of k.
It is worthwhile first to compare this problem to that of H-minor
containment: given G, test whether it has a minor isomorphic to H. For
each fixed H, this problem can be reduced to the disjoint paths
problem. The resulting algorithm will, however, have high complexity.
Similarly, the disjoint paths problem is somewhat reminiscent of the Hminor containment problem, where H consists of k independent edges.
Thus it can be solved in polynomial time.
Limitations
There are many graph problems that are believed to be not solvable in
polynomial time based on the size of the input. Examples of such
problems
include:
 Vertex Colouring: Partitioning the vertices of a graph G into a
minimum number of independent sets such that no edge joins two
vertices in the
same set.
 Traveling Salesman Problem: Finding the least cost cycle (where
costs are assigned to edges) that visits every node exactly one
(Hamiltonian Cycle).
 Vertex Cover: Find the smallest subset S of V (G) such that every
edge
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of G has at least one end in S.
However, when we restrict our input to graphs with a bound tree
width,
many of these hard problems (all of the examples) become solvable in
polynomial time.
Challenges to Practicality
We close this project with a discussion of several unusual aspects of
algorithms provided by the Graph Minor Theorem. Recall that if F is a
minor-closed family of graphs, then we know from the developments
already sketched that F can be recognized in polynomial time. Letting n
denote the number of vertices in G, the general bound is O(n3).
Interestingly, such algorithms suffer from novel shortcomings:
 the algorithms require immense constants of proportionality,
 only the complexity of decision problems is established, and
 there is no general means for finding (or even recognizing)
correct algorithms.
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Conclusions
Roberson and Seymour published a monumental body of
work that contains many theoretically and algorithmically
useful results, the vast majority of which were not covered
here. However, hopefully this project was able to provide
enough necessary background for an excited reader to feel
comfortable starting to explore their work. Anybody
interested in graph theory will likely benefit from
understanding more of their work, and there is certainly
enough breadth to keep excited along with plenty of depth
for anybody interested in details.
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