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Title: University of Florida Dept. of Computer

1
University of FloridaDept. of Computer
Information Science EngineeringCOT
3100Applications of Discrete StructuresDr.
Michael P. Frank

Slides for a Course Based on the TextDiscrete
Mathematics Its Applications (5th Edition)by
Kenneth H. Rosen

2
Module 22Graph Theory

Rosen 5th ed., chs. 8-9
44 slides (more later), 3 lectures

3
What are Graphs?
Not

General meaning in everyday math A plot or
chart of numerical data using a coordinate
system.
Technical meaning in discrete mathematicsA
particular class of discrete structures (to be
defined) that is useful for representing
relations and has a convenient webby-looking
graphical representation.

4
Applications of Graphs

Potentially anything (graphs can represent
relations, relations can describe the extension
of any predicate).
Apps in networking, scheduling, flow
optimization, circuit design, path planning.
More apps Geneology analysis, computer
game-playing, program compilation,
object-oriented design,

5
Simple Graphs

Correspond to symmetric,irreflexive binary
relations R.
A simple graph G(V,E)consists of
a set V of vertices or nodes (V corresponds to
the universe of the relation R),
a set E of edges / arcs / links unordered pairs
of distinct elements u,v ? V, such that uRv.

Visual Representationof a Simple Graph
6
Example of a Simple Graph

Let V be the set of states in the
far-southeastern U.S.
I.e., VFL, GA, AL, MS, LA, SC, TN, NC
Let Eu,vu adjoins v
FL,GA,FL,AL,FL,MS, FL,LA,GA,AL,AL,
MS, MS,LA,GA,SC,GA,TN,
SC,NC,NC,TN,MS,TN, MS,AL

NC
TN
SC
MS
AL
GA
LA
FL
7
Multigraphs

Like simple graphs, but there may be more than
one edge connecting two given nodes.
A multigraph G(V, E, f ) consists of a set V of
vertices, a set E of edges (as primitive
objects), and a functionfE?u,vu,v?V ? u?v.
E.g., nodes are cities, edgesare segments of
major highways.

Paralleledges
8
Pseudographs

Like a multigraph, but edges connecting a node to
itself are allowed. (R may be reflexive.)
A pseudograph G(V, E, f ) wherefE?u,vu,v?V
. Edge e?E is a loop if f(e)u,uu.
E.g., nodes are campsitesin a state park, edges
arehiking trails through the woods.

9
Directed Graphs

Correspond to arbitrary binary relations R, which
need not be symmetric.
A directed graph (V,E) consists of a set of
vertices V and a binary relation E on V.
E.g. V set of People,E(x,y) x loves y

10
Directed Multigraphs

Like directed graphs, but there may be more than
one arc from a node to another.
A directed multigraph G(V, E, f ) consists of a
set V of vertices, a set E of edges, and a
function fE?V?V.
E.g., Vweb pages,Ehyperlinks. The WWW isa
directed multigraph...

11
Types of Graphs Summary

Summary of the books definitions.
Keep in mind this terminology is not fully
standardized across different authors...

12
8.2 Graph Terminology

You need to learn the following terms
Adjacent, connects, endpoints, degree, initial,
terminal, in-degree, out-degree, complete,
cycles, wheels, n-cubes, bipartite, subgraph,
union.

13
Adjacency

Let G be an undirected graph with edge set E.
Let e?E be (or map to) the pair u,v. Then we
say
u, v are adjacent / neighbors / connected.
Edge e is incident with vertices u and v.
Edge e connects u and v.
Vertices u and v are endpoints of edge e.

14
Degree of a Vertex

Let G be an undirected graph, v?V a vertex.
The degree of v, deg(v), is its number of
incident edges. (Except that any self-loops are
counted twice.)
A vertex with degree 0 is called isolated.
A vertex of degree 1 is called pendant.

15
Handshaking Theorem

Let G be an undirected (simple, multi-, or
pseudo-) graph with vertex set V and edge set E.
Then
Corollary Any undirected graph has an even
number of vertices of odd degree.

16
Directed Adjacency

Let G be a directed (possibly multi-) graph, and
let e be an edge of G that is (or maps to) (u,v).
Then we say
u is adjacent to v, v is adjacent from u
e comes from u, e goes to v.
e connects u to v, e goes from u to v
the initial vertex of e is u
the terminal vertex of e is v

17
Directed Degree

Let G be a directed graph, v a vertex of G.
The in-degree of v, deg?(v), is the number of
edges going to v.
The out-degree of v, deg?(v), is the number of
edges coming from v.
The degree of v, deg(v)?deg?(v)deg?(v), is the
sum of vs in-degree and out-degree.

18
Directed Handshaking Theorem

Let G be a directed (possibly multi-) graph with
vertex set V and edge set E. Then
Note that the degree of a node is unchanged by
whether we consider its edges to be directed or
undirected.

19
Special Graph Structures

Special cases of undirected graph structures
Complete graphs Kn
Cycles Cn
Wheels Wn
n-Cubes Qn
Bipartite graphs
Complete bipartite graphs Km,n

20
Complete Graphs

For any n?N, a complete graph on n vertices, Kn,
is a simple graph with n nodes in which every
node is adjacent to every other node ?u,v?V
u?v?u,v?E.

K1
K4
K3
K2
K5
K6
Note that Kn has edges.
21
Cycles

For any n?3, a cycle on n vertices, Cn, is a
simple graph where Vv1,v2, ,vn and
Ev1,v2,v2,v3,,vn?1,vn,vn,v1.

C3
C4
C5
C6
C8
C7
How many edges are there in Cn?
22
Wheels

For any n?3, a wheel Wn, is a simple graph
obtained by taking the cycle Cn and adding one
extra vertex vhub and n extra edges vhub,v1,
vhub,v2,,vhub,vn.

W3
W4
W5
W6
W8
W7
How many edges are there in Wn?
23
n-cubes (hypercubes)

For any n?N, the hypercube Qn is a simple graph
consisting of two copies of Qn-1 connected
together at corresponding nodes. Q0 has 1 node.

Q0
Q1
Q2
Q4
Q3
Number of vertices 2n. Number of edgesExercise
to try!
24
n-cubes (hypercubes)

For any n?N, the hypercube Qn can be defined
recursively as follows
Q0v0,? (one node and no edges)
For any n?N, if Qn(V,E), where Vv1,,va and
Ee1,,eb, then Qn1(V?v1,,va,
E?e1,,eb?v1,v1,v2,v2,,va,va)
where v1,,va are new vertices, and where if
eivj,vk then eivj,vk.

25
Bipartite Graphs

Defn. A graph G(V,E) is bipartite (two-part)
iff V V1nV2 where V1?V2? and ?e?E
?v1?V1,v2?V2 ev1,v2.
In English The graph canbe divided into two
partsin such a way that all edges go between
the two parts.

V2
V1
This definition can easily be adapted for the
case of multigraphs and directed graphs as well.
Can represent withzero-one matrices.
26
Complete Bipartite Graphs

For m,n?N, the complete bipartite graph Km,n is a
bipartite graph where V1 m, V2 n, and E
v1,v2v1?V1 ? v2?V2.
That is, there are m nodes in the left part, n
nodes in the right part, and every node in the
left part is connected to every node in the
right part.

K4,3
Km,n has _____ nodesand _____ edges.
27
Subgraphs

A subgraph of a graph G(V,E) is a graph H(W,F)
where W?V and F?E.

G
H
28
Graph Unions

The union G1?G2 of two simple graphs G1(V1, E1)
and G2(V2,E2) is the simple graph (V1?V2, E1?E2).

?
29
8.3 Graph Representations Isomorphism

Graph representations
Adjacency lists.
Adjacency matrices.
Incidence matrices.
Graph isomorphism
Two graphs are isomorphic iff they are identical
except for their node names.

30
Adjacency Lists

A table with 1 row per vertex, listing its
adjacent vertices.

b
a
d
c
e
f
31
Directed Adjacency Lists

1 row per node, listing the terminal nodes of
each edge incident from that node.

32
Adjacency Matrices

A way to represent simple graphs
possibly with self-loops.
Matrix Aaij, where aij is 1 if vi, vj is an
edge of G, and is 0 otherwise.
Can extend to pseudographs by letting each matrix
elements be the number of links (possibly gt1)
between the nodes.

33
Graph Isomorphism

Formal definition
Simple graphs G1(V1, E1) and G2(V2, E2) are
isomorphic iff ? a bijection fV1?V2 such that ?
a,b?V1, a and b are adjacent in G1 iff f(a) and
f(b) are adjacent in G2.
f is the renaming function between the two node
sets that makes the two graphs identical.
This definition can easily be extended to other
types of graphs.

34
Graph Invariants under Isomorphism

Necessary but not sufficient conditions for
G1(V1, E1) to be isomorphic to G2(V2, E2)
We must have that V1V2, and E1E2.
The number of vertices with degree n is the same
in both graphs.
For every proper subgraph g of one graph, there
is a proper subgraph of the other graph that is
isomorphic to g.

35
Isomorphism Example

If isomorphic, label the 2nd graph to show the
isomorphism, else identify difference.

d
b
b
a
a
d
c
e
f
e
c
f
36
Are These Isomorphic?

If isomorphic, label the 2nd graph to show the
isomorphism, else identify difference.

Same of vertices

a
b

Same of edges

Different of verts of degree 2! (1 vs 3)

d
e
c
37
8.4 Connectivity

In an undirected graph, a path of length n from u
to v is a sequence of adjacent edges going from
vertex u to vertex v.
A path is a circuit if uv.
A path traverses the vertices along it.
A path is simple if it contains no edge more than
once.

38
Paths in Directed Graphs

Same as in undirected graphs, but the path must
go in the direction of the arrows.

39
Connectedness

An undirected graph is connected iff there is a
path between every pair of distinct vertices in
the graph.
Theorem There is a simple path between any pair
of vertices in a connected undirected graph.
Connected component connected subgraph
A cut vertex or cut edge separates 1 connected
component into 2 if removed.

40
Directed Connectedness

A directed graph is strongly connected iff there
is a directed path from a to b for any two verts
a and b.
It is weakly connected iff the underlying
undirected graph (i.e., with edge directions
removed) is connected.
Note strongly implies weakly but not vice-versa.

41
Paths Isomorphism

Note that connectedness, and the existence of a
circuit or simple circuit of length k are graph
invariants with respect to isomorphism.

42
Counting Paths w Adjacency Matrices

Let A be the adjacency matrix of graph G.
The number of paths of length k from vi to vj is
equal to (Ak)i,j.
The notation (M)i,j denotes mi,j where mi,j
M.

43
8.5 Euler Hamilton Paths

An Euler circuit in a graph G is a simple circuit
containing every edge of G.
An Euler path in G is a simple path containing
every edge of G.
A Hamilton circuit is a circuit that traverses
each vertex in G exactly once.
A Hamilton path is a path that traverses each
vertex in G exactly once.

44
Bridges of Königsberg Problem

Can we walk through town, crossing each bridge
exactly once, and return to start?

A
D
B
C
Equivalent multigraph
The original problem
45
Euler Path Theorems

Theorem A connected multigraph has an Euler
circuit iff each vertex has even degree.
Proof
(?) The circuit contributes 2 to degree of each
node.
(?) By construction using algorithm on p. 580-581
Theorem A connected multigraph has an Euler
path (but not an Euler circuit) iff it has
exactly 2 vertices of odd degree.
One is the start, the other is the end.

46
Euler Circuit Algorithm

Begin with any arbitrary node.
Construct a simple path from it till you get back
to start.
Repeat for each remaining subgraph, splicing
results back into original cycle.

47
Round-the-World Puzzle

Can we traverse all the vertices of a
dodecahedron, visiting each once?

Equivalentgraph
Dodecahedron puzzle
Pegboard version
48
Hamiltonian Path Theorems

Diracs theorem If (but not only if) G is
connected, simple, has n?3 vertices, and ?v
deg(v)?n/2, then G has a Hamilton circuit.
Ores corollary If G is connected, simple, has
n3 nodes, and deg(u)deg(v)n for every pair u,v
of non-adjacent nodes, then G has a Hamilton
circuit.

49
HAM-CIRCUIT is NP-complete

Let HAM-CIRCUIT be the problem
Given a simple graph G, does G contain a
Hamiltonian circuit?
This problem has been proven to be NP-complete!
This means, if an algorithm for solving it in
polynomial time were found, it could be used to
solve all NP problems in polynomial time.

50
8.6 Shortest-Path Problems

Not covering this semester.

51
8.7 Planar Graphs

Not covering this semester.

52
8.8 Graph Coloring

Not covering this semester.

53
9.1 Introduction to Trees

A tree is a connected undirected graph that
contains no circuits.
Theorem There is a unique simple path between
any two of its nodes.
A (not-necessarily-connected) undirected graph
without simple circuits is called a forest.
You can think of it as a set of trees having
disjoint sets of nodes.
A leaf node in a tree or forest is any pendant or
isolated vertex. An internal node is any
non-leaf vertex (thus it has degree ___ ).

54
Tree and Forest Examples
Leaves in green, internal nodes in brown.

A Tree

A Forest

55
Rooted Trees

A rooted tree is a tree in which one node has
been designated the root.
Every edge is (implicitly or explicitly) directed
away from the root.
You should know the following terms about rooted
trees
Parent, child, siblings, ancestors, descendents,
leaf, internal node, subtree.

56
Rooted Tree Examples

Note that a given unrooted tree with n nodes
yields n different rooted trees.

Same tree exceptfor choiceof root
root
root
57
Rooted-Tree Terminology Exercise

Find the parent,children, siblings,ancestors,
descendants of node f.

o
n
h
r
d
m
b
root
a
c
g
e
q
i
f
l
j
k
p
58
n-ary trees

A rooted tree is called n-ary if every vertex has
no more than n children.
It is called full if every internal (non-leaf)
vertex has exactly n children.
A 2-ary tree is called a binary tree.
These are handy for describing sequences of
yes-no decisions.
Example Comparisons in binary search algorithm.

59
Which Tree is Binary?

Theorem A given rooted tree is a binary tree iff
every node other than the root has degree ___,
and the root has degree ___.

60
Ordered Rooted Tree

This is just a rooted tree in which the children
of each internal node are ordered.
In ordered binary trees, we can define
left child, right child
left subtree, right subtree
For n-ary trees with ngt2, can use terms like
leftmost, rightmost, etc.

61
Trees as Models

Can use trees to model the following
Saturated hydrocarbons
Organizational structures
Computer file systems
In each case, would you use a rooted or a
non-rooted tree?

62
Some Tree Theorems

Any tree with n nodes has e n-1 edges.
Proof Consider removing leaves.
A full m-ary tree with i internal nodes has
nmi1 nodes, and ?(m-1)i1 leaves.
Proof There are mi children of internal nodes,
plus the root. And, ? n-i (m-1)i1. ?
Thus, when m is known and the tree is full, we
can compute all four of the values e, i, n, and
?, given any one of them.

63
Some More Tree Theorems

Definition The level of a node is the length of
the simple path from the root to the node.
The height of a tree is maximum node level.
A rooted m-ary tree with height h is called
balanced if all leaves are at levels h or h-1.
Theorem There are at most mh leaves in an m-ary
tree of height h.
Corollary An m-ary tree with ? leaves has
height h?logm?? . If m is full and balanced
then h?logm??.

64
9.2 Applications of Trees

Binary search trees
A simple data structure for sorted lists
Decision trees
Minimum comparisons in sorting algorithms
Prefix codes
Huffman coding
Game trees

65
Binary Search Trees

A representation for sorted sets of items.
Supports the following operations in T(log n)
average-case time
Searching for an existing item.
Inserting a new item, if not already present.
Supports printing out all items in T(n) time.
Note that inserting into a plain sequence ai
would instead take T(n) worst-case time.

66
Binary Search Tree Format

Items are stored at individual tree nodes.
We arrange for the tree to always obey this
invariant
For every item x,
Every node in xs left subtree is less than x.
Every node in xs right subtree is greater than x.

Example
7
3
12
1
5
9
15
0
2
8
11
67
Recursive Binary Tree Insert

procedure insert(T binary tree, x item)v
rootTif v null then begin rootT x
return Done endelse if v x return Already
presentelse if x lt v then return
insert(leftSubtreeT, x)else must be x gt
v return insert(rightSubtreeT, x)

68
Decision Trees (pp. 646-649)

A decision tree represents a decision-making
process.
Each possible decision point or situation is
represented by a node.
Each possible choice that could be made at that
decision point is represented by an edge to a
child node.
In the extended decision trees used in decision
analysis, we also include nodes that represent
random events and their outcomes.

69
Coin-Weighing Problem

Imagine you have 8 coins, oneof which is a
lighter counterfeit, and a free-beam balance.
No scale of weight markings is required for this
problem!
How many weighings are needed to guarantee that
the counterfeit coin will be found?

?
70
As a Decision-Tree Problem

In each situation, we pick two disjoint and
equal-size subsets of coins to put on the scale.

A given sequence ofweighings thus yieldsa
decision tree withbranching factor 3.
The balance thendecides whether to tip left,
tip right, or stay balanced.
71
Applying the Tree Height Theorem

The decision tree must have at least 8 leaf
nodes, since there are 8 possible outcomes.
In terms of which coin is the counterfeit one.
Recall the tree-height theorem, h?logm??.
Thus the decision tree must have heighth
?log38? ?1.893? 2.
Lets see if we solve the problem with only 2
weighings

72
General Solution Strategy

The problem is an example of searching for 1
unique particular item, from among a list of n
otherwise identical items.
Somewhat analogous to the adage of searching for
a needle in haystack.
Armed with our balance, we can attack the problem
using a divide-and-conquer strategy, like whats
done in binary search.
We want to narrow down the set of possible
locations where the desired item (coin) could be
found down from n to just 1, in a logarithmic
fashion.
Each weighing has 3 possible outcomes.
Thus, we should use it to partition the search
space into 3 pieces that are as close to
equal-sized as possible.
This strategy will lead to the minimum possible
worst-case number of weighings required.

73
General Balance Strategy

On each step, put ?n/3? of the n coins to be
searched on each side of the scale.
If the scale tips to the left, then
The lightweight fake is in the right set of ?n/3?
n/3 coins.
If the scale tips to the right, then
The lightweight fake is in the left set of ?n/3?
n/3 coins.
If the scale stays balanced, then
The fake is in the remaining set of n - 2?n/3?
n/3 coins that were not weighed!
Except if n mod 3 1 then we can do a little
better by weighing ?n/3? of the coins on each
side.