2.
Deﬁnitions of a Graph
● DEFINITION (INFORMAL): GRAPH A graph is a
nonempty set of nodes (vertices) and a set of arcs (edges) such
that each arc connects two nodes.
● Our graphs will always have a ﬁnite number of nodes and arcs.
● Example: The set of nodes in the airline map below is {Chicago,
Nashville, Miami, Dallas, St. Louis, Albuquerque, Phoenix,
Denver, San Francisco, Los Angeles}. There are 16 arcs;
Phoenix–Albuquerque, Chicago−Nashville, Miami−Dallas,
and so on.
Section 5.1 Graphs and Their Representations 2
3.
Formal Deﬁnition of a Graph
● The informal deﬁnition of a graph works quite well if
we have the visual representation of the graph before
us to show which arcs connect which nodes.
● But without the picture, we need a concise way to
convey this information.
● DEFINITION (FORMAL): GRAPH A graph is an ordered
triple (N, A, g) where:
■
N = a nonempty set of nodes (vertices)
■
A = a set of arcs (edges)
■
g = a function associating with each arc a an unordered pair
x–y of nodes called the endpoints of a
Section 5.1 Graphs and Their Representations 3
4.
Formal Deﬁnition of a Graph: Example
● Sketch a graph having nodes {1, 2, 3, 4, 5}, arcs {a1,
a2, a3, a4, a5, a6}, and function g(a1) = 1–2, g(a2) = 1–
3, g(a3) = 3–4, g(a4) = 3–4, g(a5) = 4–5, and g(a6) =
5–5.
Section 5.1 Graphs and Their Representations 4
5.
Directed Graphs
● Directed graph: the arcs of a graph begin at one node
and end at another.
● DEFINITION: DIRECTED GRAPH A directed graph
(digraph) is an ordered triple (N, A, g) where:
■
N = a nonempty set of nodes
■
A = a set of arcs
■
g = a function associating with each arc a an ordered pair (x,
y) of nodes where x is the initial point and y is the terminal
point of a
● A directed graph, therefore, has a direction associated
with each arc.
Section 5.1 Graphs and Their Representations 5
6.
Directed Graphs: Example
● In the example below there are four nodes and ﬁve
arcs.
● The function g associating arcs with endpoints
performs the mapping g(a1) = (1, 2), meaning that arc
a1 begins at node 1 and ends at node 2. Also, g(a3) =
(1, 3), but g(a4) = (3, 1).
Section 5.1 Graphs and Their Representations 6
7.
Other Forms of Graphs
● Besides imposing direction on the arcs of a graph, we may
want to modify the basic deﬁnition of a graph in other
ways.
● Labeled graph:A graph whose nodes carry identifying
information, such as the names of the cities in the map of
airline routes.
● Weighted graph: A graph where each arc has some
numerical value, or weight, associated with it. For
example, a graph that indicates the distances of the various
routes in the airline map.
● The term “graph” is used to mean an undirected graph. To
refer to a directed graph, one always says “directed graph.”
Section 5.1 Graphs and Their Representations 7
8.
Applications of Graphs
● Although the idea of a graph is very simple, an amazing number
of situations have relationships between items that lend
themselves to graphical representation.
■
Graphical representations of partially ordered sets:
• (Hasse diagrams) were introduced in Chapter 4.
■
A PERT chart (e.g., Figure 4.7) is a directed graph. The E-R
diagram (e.g., Figure 4.10) is a graph.
■
The commutative diagram illustrating composition of
functions (see Figure 4.23) is a directed graph.
■
Chapter 7 will introduce logic networks and represent them
as directed graphs.
■
Directed graphs will also be used to describe ﬁnite-state
machines in Chapter 8.
■
As seen earlier, the airline route map was a graph.
• Any representation of transportation routes is a graph (road
maps, computer networks, water pipelines, etc.).
Section 5.1 Graphs and Their Representations 8
9.
Graph Terminology
● Two nodes in a graph are adjacent if they are the endpoints associated
with an arc. 1 and 3 are adjacent nodes, but 1 and 4 are not.
● A loop in a graph is an arc with endpoints n–n for some node n; arc a3
is a loop with endpoints 2–2.
● A graph with no loops is loop-free.
● Two arcs with the same endpoints are parallel arcs; arcs a1 and a2 are
parallel.
● A simple graph is one with no loops or parallel arcs.
● An isolated node is adjacent to no other node; 5 is an isolated node.
● The degree of a node is the number of arc ends at that node.
■
Nodes 1 and 3 have degree 3, node 2 has degree 5, node 4 has degree 1,
and node 5 has degree 0.
Section 5.1 Graphs and Their Representations 9
10.
Graph Terminology
● Because the function g that relates arcs to endpoints in the
formal deﬁnition of a graph is indeed a function, each arc has a
unique pair of endpoints.
■
If g is a one-to-one function, then there is at most one arc
associated with a pair of endpoints; such graphs have no parallel
arcs.
● A complete graph is one in which any two distinct nodes are
adjacent.
● A subgraph of a graph consists of a set of nodes and a set of
arcs that are subsets of the original node set and arc set,
respectively, in which the endpoints of an arc must be the same
nodes as in the original graph.
■
Below is a complete subgraph of the graph from the previous slide.
Section 5.1 Graphs and Their Representations 10
11.
Graph Terminology
● A path from node n0 to node nk is a sequence
n0, a0, n1, a1, ... , nk − 1, ak − 1, nk
of nodes and arcs where, for each i, the endpoints of arc ai are
ni–ni + 1. If such a path exists, then nk is reachable n0.
■
In Figure seen here, one path from node 2 to node 4 consists of the
sequence 2, al, 1, a2, 2, a4, 3, a6, 4.
● The length of a path is the number of arcs it contains; if an arc
is used more than once, it is counted each time it is used.
■
The length of the path described above from node 2 to node 4 is 4.
● A graph is connected if there is a path from any node to any
other node.
● A cycle in a graph is a path from some node n0 back to n0, where
no arc appears more than once in the path sequence, n0 is the
only node appearing more than once, and n0 occurs only at the
ends.
Section 5.1 Graphs and Their Representations 11
12.
Bipartite Complete Graphs
● DEFINITION: BIPARTITE COMPLETE GRAPH A graph
is a bipartite complete graph if its nodes can be partitioned
into two disjoint nonempty sets N1 and N2 such that two nodes x
and y are adjacent if and only if x ∈ N1 and y ∈ N2. If N1 = m
and N2 = n, such a graph is denoted by Km,n.
● The ﬁgure below is not a complete graph because it is not true
that every node is adjacent to every other node.
● However, the nodes can be divided into two disjoint sets, {1, 2}
and {3, 4, 5}, such that any two nodes chosen from the same set
are not adjacent but any two nodes chosen one from each set are
adjacent.
Section 5.1 Graphs and Their Representations 12
13.
Isomorphic Graphs
● Two graphs may appear quite different in their visual
representation but still be the same graph according to our
formal deﬁnition.
● The graphs in the ﬁgures below are the same—they have
the same nodes, the same arcs, and the same arc-to-
endpoint function.
● Structures that are the same except for relabeling are called
isomorphic structures.
Section 5.1 Graphs and Their Representations 13
14.
Isomorphic Graphs
● DEFINITION: ISOMORPHIC GRAPHS Two graphs (N1, A1,
g1,) and (N2, A2, g2) are isomorphic if there are bijections f1: N1
→ N2 and f2: A1 → A2 such that for each arc a ε A1, g1(a) = x–y
if and only if g2[ f2(a)] = f1(x)–f1(y).
● The bijections for the isomorphic graphs below are:
Section 5.1 Graphs and Their Representations 14
15.
Isomorphism in Simple Graphs
● Graph isomorphism is easier to establish if we restrict our
attention to simple graphs.
● If one can ﬁnd an appropriate function f1 mapping nodes to
nodes, then a function f2 mapping arcs to arcs is trivial
because there is at most one arc between any pair of
endpoints.
● THEOREM ON SIMPLE GRAPH ISOMORPHISM
Two simple graphs (N1, A1, g1) and (N2, A2, g2) are
isomorphic if there is a bijection f: N1 → N2 such that for
any nodes ni and nj of N1, ni and nj are adjacent if and only
if f (ni) and f (nj) are adjacent. (The function f is called an
isomorphism from graph 1 to graph 2.)
Section 5.1 Graphs and Their Representations 15
16.
Proving That Graphs Are Not Isomorphic
● To prove that two graphs are not isomorphic, we must
prove that the necessary bijection(s) do(es) not exist.
● Certain conditions under which it is clear that two graphs
are not isomorphic are:
1. One graph has more nodes than the other.
2. One graph has more arcs than the other.
3. One graph has parallel arcs and the other does not.
4. One graph has a loop and the other does not.
5. One graph has a node of degree k and the other does not.
6. One graph is connected and the other is not.
7. One graph has a cycle and the other does not.
Section 5.1 Graphs and Their Representations 16
17.
Planar Graphs
● A planar graph is one that can be represented (on a sheet
of paper, that is, in the plane) so that its arcs intersect only
at nodes.
● Designers of integrated circuits want all components in one
layer of a chip to form a planar graph so that no
connections cross.
● A simple, connected, planar graph (when drawn in its
planar representation, with no arcs crossing) divides the
plane into a number of regions, including totally enclosed
regions and one inﬁnite exterior region.
● Leonhard Euler observed a relationship between the
number n of nodes, the number a of arcs, and the number r
of regions in such a graph. This relationship is known as
Euler’s formula: n − a + r = 2.
Section 5.1 Graphs and Their Representations 17
18.
Planar Graphs
● THEOREM ON THE NUMBER OF NODES
AND ARCS For a simple, connected, planar graph
with n nodes and a arcs:
1. If the planar representation divides the plane into r
regions, then n − a + r = 2.
2. If n ≥ 3, then a ≤ 3n − 6.
3. If n ≥ 3 and there are no cycles of length 3, then a ≤
2n – 4.
● We can use this theorem to prove that certain graphs
are not planar.
Section 5.1 Graphs and Their Representations 18
19.
Homeomorphic Graphs
● DEFINITION: HOMEOMORPHIC GRAPHS Two graphs
are homeomorphic if both can be obtained from the same graph
by a sequence of elementary subdivisions in which a single arc
x–y is replaced by two new arcs xv–vy connecting to a new node
v.
● The graphs in parts (b) and (c) of the ﬁgure below are
homeomorphic because each can be obtained from part (a) by a
sequence of elementary subdivisions. (However, neither can be
obtained from the other by a sequence of elementary
subdivisions.)
Section 5.1 Graphs and Their Representations 19
20.
Nonplanar Graphs
● A graph that is planar cannot be turned into a
nonplanar graph by elementary subdivisions.
● A graph that is nonplanar cannot be turned into a
planar graph by elementary subdivisions.
● Homeomorphic graphs are either both planar or both
nonplanar.
● KURATOWSKI THEOREM A graph is nonplanar
if and only if it contains a subgraph that is
homeomorphic to K5 or K3,3.
● If a graph has a subgraph homeomorphic to the
nonplanar graphs K5 or K3,3, then the subgraph—and
hence the entire graph—is nonplanar.
Section 5.1 Graphs and Their Representations 20
21.
Adjacency Matrices
● The usual computer representations of a graph involve one of two data
structures, either an adjacency matrix or an adjacency list.
● Suppose a graph has n nodes numbered n1, n2, ... , nn. This numbering
imposes an arbitrary ordering on the set of nodes; recall that a set is an
unordered collection. Having ordered the nodes, we can form an n x n
matrix where entry i,j is the number of arcs between nodes ni and nj.
This matrix is called the adjacency matrix A of the graph with respect
to this ordering. Thus, aij = p where there are p arcs between ni and nj.
● For example, the following graph has a corresponding adjacency
matrix.
Section 5.1 Graphs and Their Representations 21
22.
Adjacency Lists
● Many graphs, far from being complete graphs, have relatively few
arcs. Such graphs have sparse adjacency matrices; that is, the
adjacency matrices contain many zeros.
● Yet if the graph has n nodes, it still requires n2 data items to represent
the adjacency matrix, even if many of these items are zero. Any
algorithm or procedure in which every arc in the graph must be
examined requires looking at all n2 items in the matrix.
● A graph with relatively few arcs can be represented more
efﬁciently by storing only the nonzero entries of the adjacency
matrix.
● An adjacency list consists of a list for each node of all the
nodes adjacent to it.
■
Pointers are used to get us from one item in the list to the next.
Such an arrangement is called a linked list.
■
There is an array of n pointers, one for each node, to get each list
started.
Section 5.1 Graphs and Their Representations 22
23.
Adjacency Lists: Example
● The adjacency list for the graph of the ﬁgure on the left contains
a four-element array of pointers, one for each node.
● The pointer for each node points to an adjacent node, which
points to another adjacent node, and so forth.
● In the ﬁgure, the dot indicates a null pointer, meaning that there
is nothing more to be pointed to or that the end of the list has
been reached.
Section 5.1 Graphs and Their Representations 23
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