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
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
is a pair
is a Fnite set whose members are called
(they are dots in the above example), and
is a Fnite
set whose members are called
(they are links in the above example), where each edge
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. 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
be intervals on the real line. Then we can deFne a graph with vertex
and edge set
. Graphs arisen this way are called
which were introduced independently by mathematician Haj¨os (1957) and biologist Benzer (1959). See
GraphsDNAseq.pdf for applications of interval graphs in