Graph Theory

About the second edition
Naturally, I am delighted at having to write this addendum so soon after
this book came out in the summer of 1997. It is particularly gratifying
to hear that people are gradually adopting it not only for their personal
use but more and more also as a course text; this, after all, was my aim
when I wrote it, and my excuse for agonizing more over presentation
than I might otherwise have done.
There are two major changes. The last chapter on graph minors

now gives a complete proof of one of the major results of the Robertson-
Seymour theory, their theorem that excluding a graph as a minor bounds
the tree-width if and only if that graph is planar. This short proof did
not exist when I wrote the ¯rst edition, which is why I then included a
short proof of the next best thing, the analogous result for path-width.
That theorem has now been dropped from Chapter 12. Another addition
in this chapter is that the tree-width duality theorem, Theorem 12.3.9,
now comes with a (short) proof too.
The second major change is the addition of a complete set of hints
for the exercises. These are largely Tommy Jensen's work, and I am
grateful for the time he donated to this project. The aim of these hints
is to help those who use the book to study graph theory on their own,
but not to spoil the fun. The exercises, including hints, continue to be
intended for classroom use.
Apart from these two changes, there are a few additions. The most
noticable of these are the formal introduction of depth-¯rst search trees
in Section 1.5 (which has led to some simpli¯cations in later proofs) and
an ingenious new proof of Menger's theorem due to BÄohme, G Äoring and
Harant (which has not otherwise been published).
Finally, there is a host of small simpli¯cations and clari¯cations
of arguments that I noticed as I taught from the book, or which were
pointed out to me by others. To all these I o®er my special thanks.
The Web site for the book has followed me to