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2. To paraphrase what Felix Klein said about
curves,1 it is easy to dene a graph until you
realize the countless number of exceptions.
There are directed graphs, weighted graphs,
multigraphs, simple graphs, and so on. Where
do we begin?

3. Typical topics include:
 Introduction to graphs: undirected graphs, directed graphs,
weighted graphs, graph representation and special classes of
graphs (trees, planar graphs etc.).
 Applications of graphs (in telecommunications, networking etc.).
 Basic algorithmic techniques for graph problems: graph
traversals (DFS and BFS), topological sorting, Eular tours.
 Further algorithmic problems on graphs: minimum spanning
trees, shortest path problems, matching problems.
 Planar graphs and their properties. Eular's formula, planar
separateor theorem and their algorithmic applications.
 Further optimization problems on graphs including graph
colouring and graph questions in distributed systems.
 Discussing practical applications of graphs and efficient
algorithms for such practical problems. Approximation
algorithms and heuristic algorithms. Applications to searching in
massive graphs (e.g. page ranking); use of structural properties
and algebraic properties.

4. Learning Outcomes
 On completion of the module the student shoud be
able to:
 understand the basics of graphs, directed graphs,
weighted graphs and be able to relate them to
practical examples.
 use effectively algorithmic techniques to study basic
parameters and properties of graphs.
 design efficient algorithms for various optimisation
problems on graphs.
 use effectively techniques from graph theory to
approach practical problems in networking and
communication.

8.  Introduction
 Definitions and Examples
 Eulerian Graphs
 Hamiltonian Graphs
 Graph Connectivity
 Tree Graphs
 Graph Planarity
 Graph Embedding
 Graph Coloring

9.  AGD -- LEDA-based library of C++ classes for graph drawing.
 aiSee -- graph drawing tool that supports a variety of layout
methods.
 Graph Drawing Server -- Web-based graph drawing service.
 Graph Drawing Tools and Related Work
 Graphviz -- open source graph drawing software at AT&T.
 GDToolkit -- LEDA-based library of C++ classes for graph
drawing.
 3DCube -- tool for 3D graph drawing.
 Peterson -- software that can draw, edit and manipulate
simple graphs, examine properties of the graphs, and
demonstrate them using computer animation.
 Graph Theory Lessons -- A set of lessons (undergraduate level)
that go with the software 'Petersen'.

10.  Combinatorica
 Combinatorics, Probability, and Computing
 Discrete Applied Mathematics
 Discrete Mathematics
 Discussiones Mathematicae. Graph Theory
 Graphs and Combinatorics
 Graph Theory Journals
 Information Processing Letters
 Journal of Combinatorial Theory, Series A
 Journal of Combinatorial Theory, Series B
 Journal of Graph Theory
 Journal of Graph Algorithms and Applications
 Electronic Journal of Combinatorics
 European Journal of Combinatorics
 Theoretical Computer Science

11. Graphs are very basic objects in discrete mathematics,
and are particularly useful in modeling a wide variety of
problems. This course will have multiple objectives. First, we will
study several basic concepts in graph theory, getting a broad
introduction to the area and the different techniques used.
Next, we will study the algorithmic aspects of various graph
theoretic problems. The focus will be on understanding how
structural results in graph theory are useful in designing both
exact and approximation algorithm for various
fundamental and practical optimization problems. We will
also see how linear programming and semi-definite
programming play a key role here, providing us with a
glimpse of the rich area of combinatorial optimization.
Towards the end of the course, we will study some more
advanced (recent) results in the area. It will be interesting to
see how the various techniques we studied come together to
give these surprising new results.

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