Quiz 6 Vocab

Euler's Theorem

1. If a graph has all even vertices, it has a Euler path and circuit. 2. If there is exactly 2 odd vertices then there is a Euler path but not a Euler circuit. 3. Otherwise there is neither a path nor circuit (every Euler circuit is a Euler path, but not every Euler path is an Euler circuit)

Brute Force Method

1. Model problem with complete weighted graph 2. make list of all possible hamilton circuits 3. determine sum of weights of edges of hamilton circuits 4. minimum sum of weights is optimal solution

Fleury's Algorithm

1. pick any vertex to start 2. from that vertex pick an edge to traverse (see below for important rule) 3. darken that edge, as a reminder that you can't traverse it again 4. travel that edge, coming to the next vertex 5. repeat 2-4 until all edges have been traversed, and you are back at the starting vertex

Components

A disconnected graph is made up of pieces that are connected by themselves

Connected

A graph is connected if there is a path from any vertex to any other vertex.

Disconnected

A graph that is not connected

Sub graph

A set of vertices and edges selected from among those of th original graph

Even vertex

A vertex with an even number of edges attached to it

Odd Vertex

A vertex with an odd number of edges attached to it

Bridge

An edge that if elremoved from a connected graph would create a disconnected graph

Loops

An edge that starts and ends at the same vertex

Hamilton path

Contains each vertex only once

I am amazed by the the power of my new computer, weighted graph with 20 vertices

Doesn't make sense; it would take a supercomputer 3.8 years

Euler- theories and problems

Eulers theorum Fleurys Algorithm

Eulers theorem provides a procedure for finding Euler paths and circuits

False

Every complete graph has a Hamilton circuit and at least one Euler circuit

False

In a weighted graph, the lengths of the edges are proportional to their weights

False; lengths of edges do not need to represent real world values and are often changed to make the graph easier to read

Vertices

Finite sets of points

Complete graph

Had an edge between each pair of vertices

Trees- algorithm

Kurskals algorithm

Minimum Cost Spanning Tree

Least expensive spanning tree of all spanning trees under consideration

Degree of Vertex

Number of edges at the vertex

Kurskals algorithm

Select lowest cost edge Lowest edge on the graph does not form a circuit with the first edge Select the lowest cost edge that does not form a circuit with the previous edges Continue selecting lowest cost remaining edges

A graph has twelve vertices and fifteen edges. The graph has no loops or multiple edges.

The graph is not a tree (more edges than vertices?)

A graph has seven vertices and six edges. The graph has no loops or multiple edges. Which of the statements below applies to this​ description

The graph may or may not be a tree

Traveling Salesperson Problem

The problem of finding a Hamilton circuit in a complete, weighted graph for which the some of the weights of the edges is a minimum

edges

The start and end at vertices

Most connected graphs have many possible spanning trees

The statement is true. Each spanning tree is a skeleton holding together the connected​ graph, and for most graphs there are many such skeletons.

Hamilton- theories and problems

Traveling salesperson problem Nearest Neighbor Method

A graph can be drawn in many equivalent ways

True;Placement of vertices and the shapes of edges are unimportant

Adjacent

Two vertices in a graph are said to be adjacent when there is at least one edge connecting them

Hamilton Circuit

begins and ends at the same vertex