Homework 2
Graphs and Algorithms
Date: 17/2/2012
Due: 13/3/2012
Topic: Applications of Halls Theorem, bipartite matchings, max-ows, min-cuts, min spanning trees.
1. (5pts) Data Storage: There are n digital images. Being images of the same object they
are
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 1
Date: 23/3/2012
Scribe: Niels de Hoon
Introduction
In this lecture, we give a very brief introduction to topological methods, which have turned out to
be surprisingly useful in graph theor
Homework 3
Graphs and Algorithms
Date: 27/3/2012
Due: 27/4/2012
Topic: Probabilistic Techniques, Extremal Graphs, Topological Methods.
1. (5 pts) We proved that every planar graph G on n vertices has a separator of size 4 n,
which decomposes G into two or
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 1
Date: 7/2/2012
Scribe:
Introduction
A graph consists of vertices, and a set of edges between pairs of vertices. This seemingly basic object
turns out to be a remarkably versatile tool to m
Homework 1
Graphs and Algorithms
Date: 7/2/2012
Due: 2/3/2012
Throughout, n and m will denote the number of vertices and edges in a graph.
Problem 1. Graph Coloring:
1. (4 pts) Show that any graph with maximum degree d can be colored with d + 1 colors.
2.
Homework 4
Graphs and Algorithms
Date: 16/5/2012
Due: 8/6/2012
Topic: Linear Programming and Algorithms
1. (5pts) Use the Tutte-Berge formula to show that any 2-edge connected cubic graph G =
(V, E) (i.e., each vertex has degree 3 and min-cut has size at
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 5
Date: 2/3/2012
Scribe: Marieke Zantema
Planar Graphs
When a graph can be drawn in such a way that no two edges intersect each other, the graph is planar.
A planar graph with n vertices has
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 3
Date: 14/2/2012
Scribe: Ellen Dibbits
Halls Theorem
In the second lecture Halls Theorem is discussed, this lecture we started with proving it again in
another way.
Theorem 1 Halls Theorem
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 18
Date: 28/5/2012
Scribe: Martin Rooijackers
Scribe notes on matroid intersection
One nice feature about matroids is that a simple greedy algorithm allows to optimize over its
independent s
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 8
Date: 13/3/2012
Scribe: Sander Alewijnse
Dominating sets
In the previous lecture a probabilistic proof with alteration was presented. The following proof on
dominating sets is presented to
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 11
Date: 27/3/2012
Scribe: Annette Ficker
Regularity
Recall that e(X, Y ) refers to the amount of edges between sets X and Y .
Denition 1 Given sets A and B with the property |A| = |B| = n,
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 6
Date: 6/3/2012
Scribe: Quirijn Bouts
Planar separator algorithm
The proof of the planar separator can be found entirely in the scribe notes of lecture 5 [1] and is
therefore not included i
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 15 and 16
Date: 8/6/2012
Scribe: Reint den Toonder
Totally unimodular, denition and theorems
The solutions of homework set 3, which were discussed in lecture 15 are not included.
Denition 1
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 2
Date: 9/2/2012
Scribe: Mereke van Garderen
Introduction
There are three basic problems that will play an important role throughout this course: bipartite
matching, shortest path, and maxim
2WO08: Graphs and Algorithms
Instructor: Nikhil Bansal
1
Lecture 4
Date: 17/2/2012
Scribe: Marlous Theunissen
The maximum ow problem and its applicability
The maximum ow problem can be applied to solve a variety of problems. In this lectuer we will
consid