1 Introduction
1.1 Graphs and digraphs
Let us consider the K¨onigsberg bridges Problem (see page 85). Basically, what Euler did was to consider each
land area as an island and thus they can be represented by dots on the plane (which are labeled
A,B,C,D
).
In addition, bridges are represented by curves on the plane linking the corresponding dots. We need to
notice that the resulting diagram is not a geometric object, as dots can be placed on the plane arbitrarily
and curves can have any shape. What matters in this diagram are:
1. there are four dots representing the four land areas,
2. there are seven “linkes” representing the seven bridges, and
3. the “links” connect the corresponding dots.
Objects with these three characteristics are called graphs. To be precise, a
graph
is a pair
G
=(
V,E
), where
V
is a Fnite set whose members are called
vertices
(they are dots in the above example), and
E
is a Fnite
set whose members are called
edges
(they are links in the above example), where each edge
joins
either one
or two vertices. The following are related concepts:
incident, adjacent, loops, multiple edges, simple graphs, multigraphs, digraphs, arcs
.,
The word “graph” will mean “multigraph” in this class, unless otherwise stated (in our textbook, “graph”
means “simple graph”).
1.2 Examples
1. Street maps and distance maps (weighted graphs and digraphs).
2. Networks (computer, communication, transportation, social).
3. ±riendship graph (edges are not physical links).
4. Tournaments (see page 113).
5. Interval graphs. Let
I
1
,I
2
,...,I
n
be intervals on the real line. Then we can deFne a graph with vertex
set
{
1
,
2
,...,n
}
and edge set
{
ij
:
I
i
∩
I
j
±
=
∅}
. Graphs arisen this way are called
interval graphs
,
which were introduced independently by mathematician Haj¨os (1957) and biologist Benzer (1959). See
www.bioalgorithms.info/presentations/Ch08
GraphsDNAseq.pdf for applications of interval graphs in
biology.