Algorithm Design

Graphs
3
Our focus in this book is on problems with a discrete flavor. Just as continuous
mathematics is concerned with certain basic structures such as real numbers,
vectors, and matrices, discrete mathematics has developed basic combinatorial
structures that lie at the heart of the subiect. One of the most fundamental and
expressive of these is the graph.
The more one works with graphs, the more one tends to see them everywhere.
Thus, we begin by introducing the basic definitions surrounding
graphs, and list a spectrum of different algorithmic settings where graphs arise
naturally. We then discuss some basic algorithmic primitives for graphs, beginning
with the problem of connectivity and developing some fundamental
graph search techniques.
5.1 Basic Definitions and Applications
Reca!l from Chapter 1 that a graph G is simply a way of encoding pairwise
relationships among a set of objects: it consists of a collection V of nodes
and a collection E of edges, each of which "joins" two of the nodes. We thus
represent an edge e E E as a two-element subset of V: e = {u, v} for some
u, v E V, where we cal! u and v the ends of e.
Edges in a graph indicate a symmetric relationship between their ends.
Often we want to encode asymmetric relationships, and for this we use the
c!osely related notion of a directed graph. A directed graph G’ consists of a set
of nodes V and a set of directed edges E’. Each e’ ~ E’ is an ordered pair (a, v);
in other words, the roles of u and v are not interchangeable, and we call u the
tail of the edge and v the head. We will also say that edge e’ leaves node u and
enters node v.