Some Definitions from 9.4
• Let G be a pseudograph with vertex set V, edge set E,
and incidence mapping f. Let n be a positive integer. A
path of length n between vertex v and vertex w in G is a
sequence e1, e2, …, en of edges in E for which f(e1) = {v0,
v1}, f(e2) = {v1, v2}, …, f(en) = {vn-1,vn}, for some set of
vertices v0, v1, v2, …, vn-1, vn in V, with v0 = v and vn = w.
(For loops ei, we would have vi-1 = vi, and f(ei) would be a
singleton set)
• A circuit of length n is simply a path of length n which
ends where it started. In the above definition we simply
have v = w.
• A simple path or simple circuit is one in which there
are no repeated edges.
Connectedness in Undirected
Graphs
• We say that an undirected graph G is
connected provided…
9.7 Planar Graphs
• Definition
• Water, Electricity, Gas
Examples:
Euler’s Formula
Corollaries:
• If G is a connected planar simple graph with 𝑒 edge and 𝑣
vertices, where 𝑣 ≥ 3, then
• If a connected planar simple graph has 𝑒 edges and 𝑣
vertices with 𝑣 ≥ 3 and no circuits of length three, then
K5 and K3,3
Kuratowski’s Theorem
9.8 Graph Coloring
• Planar Dual Graph of a map
Coloring
• Define a coloring of a graph
• Define the chromatic number of a graph
The Four Color Theorem
• Appel and Haken, 1976
• If there is a counterexample, then there is a minimal
counterexample.
• A reducible configuration is a subgraph which cannot occur in a
minimal counterexample.
• Appel and Haken proved that every planar graph contains one of
1936 reducible configurations.
• The proof was constructed by a computer program.
Three colors is not enough
• Can you come up with a quick proof of that
fact?
Example:
Other Facts
• Since Appel and Haken’s proof, an O(n2)
algorithm has been discovered for 4coloring planar graphs.
• The problem of finding a 3-coloring of a
planar graph or deciding such does not
exist is NP-complete.
• The problem of finding a 4-coloring of a
general (non-planar) graph or deciding
such does not exist is NP-complete.
Computing the Chromatic Number
• Can you come up with a simple algorithm
for coloring a graph with a reasonably
small number of colors?
Applications
• Scheduling rooms, final exams, etc.
• Assigning roles in a play
• Frequency assignments for TV stations
Example
• There are three meeting rooms in the lodge where the
Royal Squid Captains hold their annual convention.
Seven meetings are scheduled. There are four officers:
The Exalted Octopus, the Revered Clam, the Mighty Sea
Bass, and the Mystic Eel. The Exalted Octopus must be
present for talks 1, 3, and 7. The Mystic Eel must attend
talks 2, 4, and 1. The Revered Clam can’t afford to miss
talk 2 or talk 5. Finally, the Mighty Sea Bass must be
present for talks 1, 4, and 6. What is the minimum
number of time slots needed in which to conduct the
meetings, so that each officer will be able to attend all
the meetings he must attend?
10.1 Introduction to Trees
• Definition: A tree is a connected undirected graph with
no simple circuits
• Theorem: An undirected graph is a tree if and only if
any two vertices are joined by a unique simple path.
Rooted Trees
• A rooted tree is a directed graph with all vertices except
one having indegree one. The exception is the root,
which has indegree zero. All other nodes are accessible
from the root via a unique path
• Canonical tree drawing is…
Tree Terms
• Node, parent, child, sibling, ancestor,
descendant
• Internal vertex, leaf
Tree Terms, Continued
• m-ary tree, binary tree
• Full m-ary tree
Ordered Trees
• An ordered tree is like a rooted tree, except that an
ordering is assigned to the children of every node, so
that the terms first child, left child, right child, etc, make
sense.
Properties of Trees
• Theorem: A tree with n vertices has n-1
edges
• Theorem: A full m-ary tree with i internal
vertices has n = mi+1 vertices
Relationships Between i, n, and l
•
•
Let i, n, and l be the number of internal
vertices, the total number of vertices, and the
number of leaves, respectively.
Theorem: In a full m-ary tree, all of the following
formulae apply:
a) i = (n – 1)/m and l = ((m – 1)n + 1)/m
b) n = mi + 1 and l = (m – 1)i + 1
c) n = (ml – 1) / (m – 1) and i = (l – 1)/(m – 1)
•
In other words, with m fixed, any two of the
attributes i, n, and l of a full m-ary tree can be
computed given the remaining attribute
Example: Suppose that someone starts a chain letter. Each
person who receives the letter is asked to send it on to four
other people. Some people do this, but others do not send
any letters. How many people have seen the letter, including
the first person, if no one receives more than one letter and if
the chain letter ends after there have been 100 people who
read it but did not send it out? How many people sent out the
letter?
Levels
• The level of a node is its distance from the
root. The root is at level 0, its children are at
level 1, their children are at level 2, etc.
• The height of a tree is the maximum of all the
levels of its nodes
• A tree of height h is balanced provided all its
leaves are either at height h or height h – 1.
Examples:
Leaves in an m-ary Tree
• Theorem: There are at most mh leaves in an
m-ary tree of height h.
• Corollary: If an m-ary tree of height h has l
leaves, then h  log m l  . If the tree is full and
balanced, then h  log m l 
10.2 Applications of Trees
Binary Search Trees
Maude
Louise
Ken
Isaac
George
Zack
Mary
Decision Trees
The Complexity of “Compare and
Swap” Sorting Algorithms
Theorem: A sorting algorithm based on binary comparisons
requires at least __________ comparisons.
Corollary: The number of comparisons used by a sorting
algorithm to sort n elements based on binary comparisons is
________________.
Prefix Codes and Huffman
Encoding
Binary code assigns a bit string to each character. Variablelength code can be used to compress a document- shorter
codes for more frequent characters. One example is a
prefix code where no code appears as the prefix of another.
Example of Huffman Coding:
'a'
'c'
'd'
'o'
'p'
'r'
's'
't'
.12
.02
.08
.14
.03
.11
.20
.30
Game Trees