1.1
CHAPTERl
ELEMENTS OF GRAPH
THEORY
GRAPH MODELS
The first four chapters deal with graphs and their applications. A graph G
=
(V, E)
consists of a finite set
V
of vertices and a set
E
of edges joining different pairs of
distinct vertices.
*
Figure 1.1
a
shows a depiction of a graph with
V
=
{a, b, c, d}
and
E
=
((a, b), (a, c), (a, d), (b, d), (c, d)}.
We represent vertices with points and edges
with lines joining the prescribed pairs of vertices. This definition of a graph does not
allow two edges to join the same two vertices. Also an edge cannot "loop" so that
both ends terminate at the same vertex-an edge's end vertices must be distinct. The
two ends of an undirected edge can be written in either order,
(b, c)
or
(c, b).
We say
that vertices
a
and
b
are adjacent when there is an edge
(a, b).
Sometimes the edges are ordered pairs of vertices, called directed edges. In a
directed graph, all edges are directed. See the directed graph in Figure 1.1
b.
We write
(b~
c)
to denote a directed edge from
b
to
c.
In a directed graph, we allow one edge in
each direction between a pair of vertices. See edges
(a
~
c)
and
(c~
a)
in Figure
l.lb.
The combinatorial reasoning required in graph theory, and later in the enumera-
tion part of this book, involves different types of analysis than used in calculus and
high school mathematics. There are few general rules or formulas for solving these
problems. Instead, each question usually requires its own particular analysis. This
analysis sometimes calls for clever model building or creative thinking but more of-
ten consists of breaking the problem into many cases (and subcases) that are easy
enough to solve with simple logic or basic counting rules. A related line of reason-
ing is to solve a special case of the given problem and then to find ways to extend
that reasoning to all the other cases that may arise. In graph theory, combinatorial
arguments are made a little easier by the use of pictures of the graphs. For example,
a case-by-case argument is much easier to construct when one can draw a graphical
depiction of each case.
Graphs have proven to be an extremely useful tool for analyzing situations in-
volving a set of elements in which various pairs of elements are related by some
property. The most obvious examples of graphs are sets with physical links, such
'What this book calls a graph is referred to in many graph theory books as a
simple graph.
In general,
graph theory terminology varies a little from book to book.
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