MATH 118 Test One

Total number of hamilton circuits=

(n-1)!

Euler Circuit

- Can start at any vertex - no odd vertices

Euler Path

- must start and end at an odd vertex - no more than two odd vertices

Important properties of trees

- there is only one path connecting a given pair of vertices - every edge is a bridge - a tree with n vertices has exactly n-1 edges

A complete graph with more than 2 vertices will always have .... Euler Paths

0

A complete graph with more than 2 vertices will always have

0 bridges

Trees

A connected graph that has no circuits.

Networks

A connected graph.

Unicursal Training

A graph that can be drawn without picking upyour pen and without retracing any part of the graph.

Complete Graph

A graph with exactly one edge between each pair of vertices.

Weighted Graph

A number assigned to an edge of a graph as a way of representing a cost, distance, time, etc. associated with traveling along that edge.

Which of the following statements about a connected graph is always true?

A path of edges exists between any two vertices of the graph.

Optimal Exhaustive Route

A route that travels each edge of the graph at least once by using the fewest repeated edges.

Exhaustive Route

A route that travels each edge of the graph at least once.

Hamilton Circuit

A route that visits each vertex of a connected graph exactly once, starting and ending at the same vertex.

Hamilton Path

A route that visits each vertex of a connected graph exactly once. The route is not required to start and end at the same vertex.

Minimum Spanning Tree

A spanning tree with the least total weight.

Spanning Trees

A tree that connects all vertices of the network.

Bridge

An edge whose removal makes a connected graph disconnected

When determining the amount of possible spanning trees on a graph, what do you do?

Count all of the edges of circuits on the graph (for individual circuits) and multiply them - circuit edges #1 * circuit edges #2= possible spanning trees

Closed Unicursal Tracing

Euler circuit

Open Unicursal Tracing

Euler path

Every graph with an Euler circuit has an even number of edges.

FALSE

Every graph with an Euler circuit has an even number of vertices.

FALSE

Brute Force Algorithm

List all possible Hamilton circuits for G and then choose the circuit with the smallest total weight

A network with m edges and n vertices has redundancy

R = m − n +1.

Kruskal's Algorithm

Step One: Build minimum spanning tree one edge at a time, choosing at each step the cheapest available edge Step Two: Never create a circuit - it is okay to have more than three vertices at a point - always gives the optimal solution

Cheapest Link Algorithm

Step One: Choose edge with the smallest weight Step Two: Choose the next edge with the smallest weight Step Three: Continue to choose smallest weights (avoid creating circuits before you are done and creating an odd vertex) Step Four: Connect the graphs

Fleury's Algorithm

Step One: Determine if it is a Euler circuit or path Step Two: Euler circuit can start at any vertex, while Euler Path must start and end at an odd vertex Step Three: Follow the edges one at a time, always choose the non bridge edge first Step Four: Stop when you run out of edges

Nearest-Neighbor Algorithm

Step One: Start at designated point Step Two: Go to neighboring edge with the smallest weight Step Three: Keep choosing the nearest neighbor, unless it is a repeat, until you run out of edges

Every graph that has an Euler circuit is connected.

TRUE

Redundancy

The number of duplicate edges in a network.

Eulerization

The process of adding duplicate edges to a graph so that the resulting graph has an Euler Circuit. to make all even vertices

Semi-Eulerization

The process of adding duplicate edges to a graph so that the resulting graph has an Euler Path. with two odd vertices

Circuit

a closed trip on a graph, starting and ending at the same vertex.

In order for a connected graph to have an Euler Circuit

all vertices must be even.

Path

an open trip on a graph, starting and ending at different vertices.

Every edge on a tree is a

bridge

Traveling Sales Man Problem

determine an optimal circuit that visits each vertex exactly once.

f a complete graph has n vertices

each vertex must have degree = n - 1.

In order for a connected graph to have an Euler Path

exactly 2 vertices must be odd.

Every graph with an Euler path must have at least 2 odd vertices.

false

The Cheapest Link algorithm for solving the traveling salesman problem alwaysgives optimal results.

false

The Nearest-Neighbor algorithm will never use the most expensive edge on the graph.

false

Connected Graph

if you can get from any vertex to any other vertex along a path.

f a graph has an Euler Path

it cannot have an Euler Circuit.

If a graph has an Euler Circuit

it cannot have an Euler Path.

If a graph has a vertex of degree 1

it cannot have any type of circuit.

If a graph has a Hamilton Circuit

it will also have a Hamilton Path.

A tree with n vertices has exactly

n - 1 edges.

A graph always has an even number of vertices with

odd degree.

Optimal Tour

the Hamilton circuit of least total cost.

Degree of a Vertex

the number of edges that meet at a vertex

Every graph with an Euler circuit has only vertices with even degree.

true

It is possible for a graph to have an Euler circuit, but not a Hamilton circuit.

true

The Brute Force algorithm always yields the optimal Hamilton Circuit.

true

t is possible for a graph to have a Hamilton circuit, but not an Euler circuit.

true

Graph Models

vertices represent different objects edges represent how the objects are related

In order to trace an open diagram with n odd vertices

you must lift your pencil at least n − 2/2 times.

In order to trace a closed diagram with n odd vertices

you must lift your pencil at least n/2 times.