Module 22-graphs

Module #22 - Graphs

University of Florida
Dept. of Computer & Information Science & Engineering

COT 3100
Applications of Discrete Structures
Dr. Michael P. Frank

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

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Module #22 - Graphs

Module #22:
Graph Theory

Rosen 5th ed., chs. 8-9
~44 slides

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Module #22 - Graphs

What are Graphs? Not

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

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Module #22 - Graphs

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, …
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Module #22 - Graphs

Simple Graphs
• Correspond to symmetric,
irreflexive binary relations R.
• A simple graph G=(V,E)
Visual Representation
consists of: of a Simple Graph
– 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.

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Module #22 - Graphs

Example of a Simple Graph
• Let V be the set of states in the far-
southeastern U.S.:
–I.e., V={FL, GA, AL, MS, LA, SC, TN, NC}
• Let E={{u,v}|u adjoins v}
TN NC
={{FL,GA},{FL,AL},{FL,MS},
{FL,LA},{GA,AL},{AL,MS}, MS AL SC
{MS,LA},{GA,SC},{GA,TN},
GA
{SC,NC},{NC,TN},{MS,TN}, LA
FL
{MS,AL}}
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Module #22 - Graphs

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 function Parallel
f:E→{{u,v}|u,v∈V ∧ u≠v}. edges

• E.g., nodes are cities, edges
are segments of major highways.
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Module #22 - Graphs

Pseudographs
• Like a multigraph, but edges connecting a node to
itself are allowed. (R may be reflexive.)
• A pseudograph G=(V, E, f ) where
f:E→{{u,v}|u,v∈V}. Edge e∈E is a loop if
f(e)={u,u}={u}.
• E.g., nodes are campsites
in a state park, edges are
hiking trails through the
woods.

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Module #22 - Graphs

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}

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Module #22 - Graphs

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 f:E→V×V.
• E.g., V=web pages,
E=hyperlinks. The WWW is
a directed multigraph...

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Module #22 - Graphs

Types of Graphs: Summary
• Summary of the book’s definitions.
• Keep in mind this terminology is not fully
standardized across different authors...
Edge Multiple Self-
Term type edges ok? loops ok?
Simple graph Undir. No No
Multigraph Undir. Yes No
Pseudograph Undir. Yes Yes
Directed graph Directed No Yes
Directed multigraph Directed Yes Yes

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Module #22 - Graphs

§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.

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Module #22 - Graphs

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.

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Module #22 - Graphs

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.

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Module #22 - Graphs

Handshaking Theorem
• Let G be an undirected (simple, multi-, or
pseudo-) graph with vertex set V and edge
set E. Then

∑ deg(v) = 2 E
v∈V
• Corollary: Any undirected graph has an
even number of vertices of odd degree.

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Module #22 - Graphs

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

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Module #22 - Graphs

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 v’s in-degree and out-degree.

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Module #22 - Graphs

Directed Handshaking Theorem
• Let G be a directed (possibly multi-) graph
with vertex set V and edge set E. Then:
1
∑ δεγ (v) = ∑ δεγ (v) = 2 ∑ δεγ( v) = E
v∈V
−

v∈V
+

v∈V
• Note that the degree of a node is unchanged
by whether we consider its edges to be
directed or undirected.

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Module #22 - Graphs

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
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Module #22 - Graphs

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 K2 K3 K4
K5 K6
ν( ν − 1)
ν −1

Note that Kn has ∑ ι = 2 edges.
ι=1

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Module #22 - Graphs

Cycles
• For any n≥3, a cycle on n vertices, Cn, is a
simple graph where V={v1,v2,… ,vn} and
E={{v1,v2},{v2,v3},…,{vn−1,vn},{vn,v1}}.

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

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Module #22 - Graphs

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?
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Module #22 - Graphs

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 edges:Exercise to try!

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Module #22 - Graphs

n-cubes (hypercubes)
• For any n∈N, the hypercube Qn can be
defined recursively as follows:
– Q0={{v0},∅} (one node and no edges)
– For any n∈N, if Qn=(V,E), where V={v1,…,va}
and E={e1,…,eb}, then Qn+1=(V∪{v1´,…,va´},
E∪{e1´,…,eb´}∪{{v1,v1´},{v2,v2´},…,
{va,va´}}) where v1´,…,va´ are new vertices, and
where if ei={vj,vk} then ei´={vj´,vk´}.

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Module #22 - Graphs

Bipartite Graphs
• Def’n.: A graph G=(V,E) is bipartite (two-
part) iff V = V1∩V2 where V1∪V2=∅ and
∀e∈E: ∃v1∈V1,v2∈V2: e={v1,v2}.
• In English: The graph can
be divided into two parts
in such a way that all edges
go between the two parts. V1 V2
This definition can easily be adapted for the
case of multigraphs and directed graphs as well. Can represent with
zero-one matrices.
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Module #22 - Graphs

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,v2}|v1∈V1 ∧ v2∈V2}.
– That is, there are m nodes K4,3
in the left part, n nodes in
the right part, and every
node in the left part is
connected to every node Km,n has _____ nodes
in the right part. and _____ edges.

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Module #22 - Graphs

Subgraphs
• A subgraph of a graph G=(V,E) is a graph
H=(W,F) where W⊆V and F⊆E.

G H

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Module #22 - Graphs

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).

∪
a b c a b c

d e d f

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Module #22 - Graphs

§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.

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Module #22 - Graphs

Adjacency Lists
• A table with 1 row per vertex, listing its
adjacent vertices. Adjacent
a b Vertex Vertices
a b, c
c d b a, c, e, f
e
f c a, b, f
d
e b
f c, b

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Module #22 - Graphs

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

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Module #22 - Graphs

Adjacency Matrices
• A way to represent simple graphs
– possibly with self-loops.
• Matrix A=[aij], 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 >1) between the nodes.

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Module #22 - Graphs

Graph Isomorphism
• Formal definition:
– Simple graphs G1=(V1, E1) and G2=(V2, E2) are
isomorphic iff ∃ a bijection f:V1→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.
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Module #22 - Graphs

Graph Invariants under Isomorphism

Necessary but not sufficient conditions for
G1=(V1, E1) to be isomorphic to G2=(V2, E2):
– We must have that |V1|=|V2|, and |E1|=|E2|.
– 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.

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Module #22 - Graphs

Isomorphism Example
• If isomorphic, label the 2nd graph to show
the isomorphism, else identify difference.
b d
a b a
d c
e
e c f
f

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Module #22 - Graphs

Are These Isomorphic?
• If isomorphic, label the 2nd graph to show
the isomorphism, else identify difference.
• Same # of
a vertices
b • Same # of
edges
• Different # of
d verts of
c e degree 2!
(1 vs 3)

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Module #22 - Graphs

§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 u=v.
• A path traverses the vertices along it.
• A path is simple if it contains no edge more
than once.

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Module #22 - Graphs

Paths in Directed Graphs
• Same as in undirected graphs, but the path
must go in the direction of the arrows.

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Module #22 - Graphs

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 P. Frank 39
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removed.

Module #22 - Graphs

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.
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Module #22 - Graphs

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

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Module #22 - Graphs

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.

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Module #22 - Graphs

§8.5: Euler & Hamilton Paths
• An Euler circuit in a graph G is a circuit
containing every edge of G.
• An Euler path in G is a 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.

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Module #22 - Graphs

Bridges of Königsberg Problem
• Can we walk through town, crossing each
bridge exactly once, and return to start?
A

B D

C
Equivalent multigraph
The original problem

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Module #22 - Graphs

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.

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Module #22 - Graphs

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.

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Module #22 - Graphs

Round-the-World Puzzle
• Can we traverse all the vertices of a
dodecahedron, visiting each once?`

Equivalent
Dodecahedron puzzle Pegboard version
graph

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Module #22 - Graphs

Hamiltonian Path Theorems
• Dirac’s 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.
– Ore’s corollary: If G is connected, simple, has
n≥3 nodes, and deg(u)+deg(v)≥n for every pair
u,v of non-adjacent nodes, then G has a
Hamilton circuit.

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Module #22 - Graphs

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.

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Module #22 - Graphs

§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 ≥ ___ ).
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Module #22 - Graphs

Tree and Forest Examples
Leaves in green, internal nodes in brown.
• A Tree: • A Forest:

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Module #22 - Graphs

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.

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Module #22 - Graphs

Rooted Tree Examples
• Note that a given unrooted tree with n
nodes yields n different rooted trees.
Same tree
except
root for choice
of root
root

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Module #22 - Graphs

Rooted-Tree Terminology Exercise

• Find the parent, o h n r
children, siblings,
ancestors, & d
m
b
descendants root
of node f. a c g
e
i q
f
p j k l

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Module #22 - Graphs

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.

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Module #22 - Graphs

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 ≤ ___.

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Module #22 - Graphs

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 n>2, can use terms like
“leftmost”, “rightmost,” etc.

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Module #22 - Graphs

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?

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Module #22 - Graphs

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 n=mi+1
nodes, and =(m−1)i+1 leaves.
– Proof: There are mi children of internal nodes,
plus the root. And,  = n−i = (m−1)i+1. □
• 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.

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Module #22 - Graphs

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.

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Module #22 - Graphs

§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

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Module #22 - Graphs

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

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Module #22 - Graphs

Binary Search Tree Format
• Items are stored at Example:
individual tree nodes. 7
• We arrange for the
tree to always obey 3 12
this invariant:
– For every item x,
• Every node in x’s left
1 5 9 15
subtree is less than x.
• Every node in x’s right
subtree is greater than x. 0 2 8 11

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Module #22 - Graphs

Recursive Binary Tree Insert
procedure insert(T: binary tree, x: item)
v := root[T]
if v = null then begin
root[T] := x; return “Done” end
else if v = x return “Already present”
else if x < v then
return insert(leftSubtree[T], x)
else {must be x > v}
return insert(rightSubtree[T], x)

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Module #22 - Graphs

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.

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Module #22 - Graphs

Coin-Weighing Problem
• Imagine you have 8 coins, one
of 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?

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Module #22 - Graphs

As a Decision-Tree Problem
• In each situation, we pick two disjoint and equal-
size subsets of coins to put on the scale.
The balance then A given sequence of
“decides” whether weighings thus yields
to tip left, tip right, a decision tree with
or stay balanced. branching factor 3.

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Module #22 - Graphs

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 height
h ≥ log38 = 1.893… = 2.
• Let’s see if we solve the problem with only
2 weighings…

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Module #22 - Graphs

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 what’s 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.

11/06/12 (c)2001-2003, Michael P. Frank 69

Module #22 - Graphs

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.
You can prove that this strategy always leads to a balanced 3-ary tree.
11/06/12 (c)2001-2003, Michael P. Frank 70

Module #22 - Graphs

Coin Balancing Decision Tree
• Here’s what the tree looks like in our case:

left: 123 vs 456
123 balanced:
right: 78
456

1 vs. 2 4 vs. 5 7 vs. 8
L:1 R:2 B:3 L:4 R:5 B:6 L:7 R:8

11/06/12 (c)2001-2003, Michael P. Frank 71

Module #22 - Graphs

§9.3: Tree Traversal
• Universal address systems
• Traversal algorithms
– Depth-first traversal:
• Preorder traversal
• Inorder traversal
• Postorder traversal
– Breadth-first traversal
• Infix/prefix/postfix notation

11/06/12 (c)2001-2003, Michael P. Frank 72

Module 22-graphs

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