4.1 Introduction to Graphs
• Graph – consists of vertices and edges.
Each edge must start and end at a vertex.
• Example:
4.1 Introduction to Graphs
• Examples of non-graphs:
(edge splits in two pieces, no vertex at end)
4.1 Introduction to Graphs
• Path – connects one vertex to another by a
sequence of edges
V1
• Example: V1- V3- V2- V4
V2
V4
V3
V5
4.1 Introduction to Graphs
• Connected Graph – there is a path from
every vertex to every other vertex
• Example of graph that is not connected:
V1
V2
V4
V3
V6
V5
4.1 Introduction to Graphs
• Degree of a vertex - number of edges
connected to it
• Example: V4 has degree = 3
V2
V4
V1
V3
V5
4.1 Introduction to Graphs
• Total degree of a graph - sum of the degrees of all
the vertices.
Note: If a graph has n edges, the total degree = 2n.
• Example: graph has total degree = 10
V1
V4
V2
V3
4.2 Edge Paths
• Konigsberg Bridge Problem: Can you walk
across each bridge exactly once?
North Bank
River
Island
Island
South Bank
4.2 Edge Paths
• In 1736, Euler showed that the Konigsberg bridge
problem was unsolvable by using graph theory.
• Another example – trace without lifting your
pencil:
4.2 Edge Paths
• Edge Path – a path that goes through each edge of
the graph exactly once
To determine if a graph has an edge path, count
the number of vertices of odd degree. Note that
this number will always be even (0,2,4,…)
• A graph has an edge path if it has:
– 2 vertices of odd degree
4.2 Edge Paths
• Selecting the starting/ending vertices:
Given 2 vertices of odd degree – start at one
of the vertices and end at the other
4.2 Fleury’s Method for Finding
an Edge Path
1. Select a starting vertex.
2. Select and number a connecting edge if it
is the only remaining unused edge, or,
select it if the edge does not cut off or
isolate a portion of the graph.
3. Repeat step 2 until all edges have been
selected.
4.3 Edge Cycles
• Cycle – a path that starts and ends on the
same vertex with no repeated edges
• Edge Cycle – a cycle that includes every
edge in the graph
• Theorem: A graph has an edge cycle  it is
connected and every vertex is of even
degree
4.3 Edge cycles
• Selecting the starting/ending vertices for
edge cycles:
– no vertices of odd degree – start at any
vertex and end at the same vertex
4.3 Edge Cycles
• Edge cycles are different from edge paths:
– Edge Cycle – every vertex is of even
degree
– Edge Path – exactly 2 vertices are of odd
degree
4.3 Edge Cycles
• Eulerizing – adding edges where they already
exist to create an edge cycle
• Method – add edges to connect the vertices of odd
degree. This changes both vertices to an even
degree. Make sure you only connect one edge to
each odd vertex!
• Note: Eulerizing is useful in museums, gardens,
street systems with non-Euler cycles that force
you to retrace your path
4.4 Vertex Cycles
• Cycle – a path that starts and ends on the
same vertex with no repeated edges
• Vertex Cycle – a cycle that contains every
vertex
• Unsolved problem: Identifying exactly
when a vertex cycle exists.
4.4 Vertex Cycles
• Dirac’s Theorem – if the following are all true for
a graph then a vertex cycle must exist:
– Graph is connected
– Graph has 3 or more vertices
– Each vertex is adjacent to at least half of the
vertices
• Note: Dirac’s conditions may not be satisfied and
yet a vertex cycle still exists.
4.4 Vertex Cycles
• Complete Graph – every vertex is
connected to every other vertex
• Definition: n! n  (n 1)  (n  2) ... 3 2 1
• Theorem: A complete graph with n vertices
has (n-1)! vertex cycles
(n  1)!
• Note: This becomes
if reverse cycles
2
are not included.
4.5 Traveling Salesman Problem
• Not the “how to get rid of a traveling salesman”
problem
• Problem: How to schedule visits to cities in the
cheapest way possible – using graph theory
• Weighted Graph – a weight (number) is assigned
to each edge
• Graph theory problem: Finding the vertex cycle
with the smallest total weight
4.5 Traveling Salesman Problem
• Solution – Brute Force Method:
– List all vertex cycles (reverse cycles not
included)
– Find the total weight of each cycle
– Pick the cycle with the smallest weight
(n  1)!
2
• Problem with this approach:
cycles
must be checked – gets large in a hurry
4.5 Traveling Salesman Problem
• Lower Bound – Add the weights of the n
shortest edges where n = # of vertices
• Applications of traveling salesman problem:
– Deliveries (e.g. Federal Express)
– Collections(e.g. Trash Pickup)
– Manufacturing(drilling holes in circuit boards –
need to minimize the movement of the drill)
4.6 Approximations to Traveling
Salesman Problem
•
Nearest Neighbor Approximation:
1. Select the starting vertex.
2. Select the unused connecting edge with the
smallest weight.
3. Repeat step 2 until all vertices are reached.
4. Select the edge that goes back to the starting
point to complete the cycle
4.6 Approximations to Traveling
Salesman Problem
•
Cheapest Link Approximation:
1. Select the edge with the smallest weight.
2. Select the unused edge with the smallest
weight unless it completes a cycle or one
vertex becomes connected to 3 edges. Break
ties arbitrarily.
3. Repeat step 2 until all vertices have been
included in a cycle.
4.6 Approximations to Traveling
Salesman Problem - Comparison
•
Nearest Neighbor Approximation:
1. Starting vertex must be specified.
2. Order selected is always sequential – starting
vertex is always the end vertex of the
previous edge.
•
Cheapest Link Approximation:
1. Order selected is not necessarily sequential.
2. Weights are always increasing.
4.7 Trees
• Tree – a graph that is connected and has no cycles.
Examples: family tree, organizational chart
• Properties of trees:
– For any 2 vertices there is a unique path
between them
– A tree with n vertices has n-1 edges
– A tree has the least number of edges possible
for a connected graph
4.7 Trees
• Determining whether a graph is a tree - A
graph with n vertices and n-1 edges must be
a tree
• Spanning Tree of a Graph – a tree that is a
subgraph that contains all the vertices of
the graph
• Minimal Spanning Tree – the spanning tree
of a graph with the smallest weight
4.7 Trees
•
Kruskal’s method for finding the minimal
spanning tree:
1. Find the edge with the smallest weight
2. Find the next smallest unused edge that
does not form a cycle
3. Repeat step 2 until all vertices are
included. Remember - do not complete
a cycle!
4.8 Instant Insanity
• Instant Insanity - uses 4 blocks with colors
on the sides (red, white, blue, green).
• Goal – to stack the cubes so that all 4 colors
occur exactly once on each side.
• There are 331,776 possibilities with trial
and error.
• The puzzle can be solved using graph
theory.
4.8 Instant Insanity
• Cubes must be represented in 2 dimensions
• Convert each cube to a graph representation
by connecting colors of opposite sides
R
B
G
W
R
W
B
G
W
G
4.8 Instant Insanity
• Consolidate to one graph, labeling
paths by cube number
2 R
R
B
WR
GB
WR
G B
WR
W
GB
2
3
4
G
1
B
W
3
1
2
1
4
G 4
3
4.8 Instant Insanity
• Solution to consolidated graph – pick 2
subgraphs such that
– Each vertex has degree 2
– Each cube is represented exactly once in each
subgraph (1, 2, 3, and 4)
– 2 subgraphs have no edge in common (same
number/edge in both subgraphs)
• Note: forms of solutions are given in book
4.8 Instant Insanity
• Solution subgraphs for the example:
2 R
R
W
R
1 2
4
B
W
1
2 4
3
G
3
B
1
34 2
W
3
1
2
1
4
G
B
G 4
3