CSCC Math 1116 Part 2: Graph Ch. 2, 3, 4

A tree with N vertices will have _______ edges.

(N-1)

A connected graph with ___ vertices and (N-1) edges will be a tree.

N

If we have 4, 6, 8, etc. ODD vertices, is there a Euler Path? What about a Circuit?

No path; No circuit

How do we find the number of possible circuits in a complete graph?

...The number of possible circuits in a graph is (n-1)!/2, where n is the number of vertices.

What is a complete graph? Give an example?

.A graph where every vertice is connected to every other vertice.

How do you navigate through the Repeated Nearest Neighbor Algorithm?

1. Do the Nearest Neighbor Algorithm starting at each vertex (start with A in first attempt at algorithm, next start with B then C, etc.) 2. Choose the circuit produced with minimal total weight

How do you navigate through the Brute Force Algorithm?

1. List all possible Hamiltonian Circuits 2. Find the cost of each circuit by adding the edge weights 3. Select the circuit with minimal total weight

How do you navigate through the Nearest Neighbor Algorithm?

1. Select a starting point 2. Move to the nearest unvisited vertex (use the edge with smallest weight) 3. Repeat until the circuit is complete

How do you navigate through the Sorted Edges Algorithm (aka Cheapest Link Algorithm)?

1. Select the cheapest unused edge in the graph 2. Repeat step 1, adding the cheapest unused edge to the circuit, unless: a. adding the edge would create a circuit that doesn't contain all vertices b. adding the edge would give a vertex degree of three. 3. Repeat until a circuit containing all vertices is formed.

What is the term for a graph that has each vertex connected to all other vertices in the graph? (In graph with ABCD, A has an edge going to B, C, and D and likewise with all other vertices reaching all others.)

A COMPLETE graph

What is a Hamiltonian Circuit?

A circuit that visits every vertex once with no repeats. Being a circuit, it must start and end at the same vertex.

What is the rule for determining if a graph has a Euler Path, according to Euler's Path and Circuit Theorem?

A graph has a Euler Path if there are exactly 0 or 2 vertices with a ODD degree... if there are exactly 2, the path will start at one and end at the other. (If there are more than 2 odd degree vertices, there is no path.)

What is a Hamiltonian Path?

A path that visits every vertex once with no repeats, but does not have to start and end at the same vertex.

What is a Path?

A series of connected vertices (no repeated edges), doesn't have to return to starting vertex

What is a Circuit?

A series of vertices that begin and end at the same vertex (no repeated edges)

A graph where there is a path from any vertex to any other vertex, but has no circuits.

A tree

What is Eulerization?

The process of duplicating edges to eliminate all odd vertices to make a Euler CIRCUIT- aka "go along the same edge twice".

What is the rule to determining a connected vs. disconnected graph?

An edge is connected by at least 2 vertices

All Hamiltonian Circuit Algorithms are heuristic, except the ____ ________ algorithm.

Brute Force

The formula for finding the number of circuits in a graph ((n-1)! / 2) only works for ___________ graphs.

Complete

The following is an example of what Theorem?; A series of rules allowing one to determine if a graph has a Euler Path, Euler Circuit, or both.

Euler's Path and Circuit Theorem

How do you use Fleury's Algorithm to find a path or circuit?

First, determine if the graph is a path or a circuit. Next, start at the odd vertex if a path, or any vertex if it is a circuit. Then delete edges that are NOT BRIDGES, one by one until the path or circuit is determined.

What is the rule for determining if a graph has a Euler Circuit, according to Euler's Path and Circuit Theorem?

If all vertices have an even degree, the path has a Euler Circuit. (If there are ANY odd degrEe vertices, there is no circuit.)

What is the difference between the Nearest Neighbor Algorithm and the Sorted Edges/ Cheapest Link Algorithm?

In the Nearest Neighbor algorithm, you must travel from vertex to vertex. In the Sorted Edges algorithm, you can select edge AC and then next select edge BD- there does not need to be a path from vertex to vertex. You are selecting and sorting the cheapest edges... order follows in selection of cheapest to 2nd cheapest to 3rd cheapest, etc. whereas the NNA, order follows from cheapest edge vertex to next cheapest edge vertex and they must be connected edges/ vertices.

Kruskal's Algorithm works similarly to which HC algorithm? (except this time you are avoiding creating a circuit).

It works similarly to Sorted Edges Algorithm, where you pick randomly the cheapest edges and create a path, except ones that create circuits. (degree also doesn't matter)

When creating a spanning tree, any old spanning tree wont do. We want to find the _________ _________ spanning tree.

Minimum cost

Is it always possible to use every edge in a circuit?

No

If we have 1, 3, 5, etc. odd vertices, is there a Euler Path? What about a Circuit?

No; No; remember, this is "Impossible"!**

When utilizing Eulerization and Semi-eulerization, it is important to be efficient, duplicating the fewest number of edges as possible. This is called what?

Optimiazation (or optimal)

How do you find an MCST?

Start with the lowest cost edge; then pick the next cheapest edge assuming adding that edge wont create a circuit. (degree of vertices doesnt matter)

An example of an inefficient but optimal algorithm is?

The Brute Force Algorithm

What is the unique/ "classic" problem that attempts to visit every vertex once using the cheapest route? (aka the most optimal route)

The Travelling Saleshuman Problem

What is Semi-eulerization? How do we semi-eulerize a graph?

The process of duplicating edges OPTIMALLY on all but 2 odd vertices to obtain a Euler PATH.

Why is it impossible for a graph to have an odd sum of degrees of vertices?

The sum of degrees is always 2M, where M is the number of edges, and any number of edges, even or odd, times 2 (2M) will be an even number, never an odd number. (2x3, 2x5, 2x7 all = even numbers, not odd.) So having an odd number of sum of degrees of vertices is impossible.

What is Fleury's Algorithm?

This Algorithm allows us to find a graph's Euler path or circuit.

What is Euler's Sum of Degree Theorem?

This theorem states that the sum of degrees of vertices of a graph is 2x the number of edges. (The equation is, "if M is the number of edges, the sum of degrees of vertices is 2M.)

What is the underlying goal while duplicating edges to create a EP or EC?

To be optimal!

T or F; Kruskal's algorithm is both optimal and efficient?

True

T or F; The Sorted Edge Algorithm is a heuristic and therefor will not guarantee an optimal circuit?

True

True or false; the degree of vertices in the Sorted Edges algorithm does matter, and we are trying not to create any circuits.

True

True or false; the heuristic graphs require complete graphs?

True

True or false; the two rules of a spanning tree are that the number of edges are N-1, and there can be no circuits?

True

True or false; when creating spanning trees, it is okay to have vertices with degree greater than 3, and we do not want any circuits?

True

True or false; when creating spanning trees, the goal is to be optimal?

True

T or F; both HC and HP visit every vertex once with no repeats?

True; except a HP doesn't have to start and end at the same vertex and a HC does.

What is a Euler Path?

Used ALL edges in a graph (no repeated edges), doesn't have to return to starting vertex

What is a Euler Circuit?

Uses ALL edges in a graph (no repeated edges), MUST start and end at the same vertex

If we have 2 ODD vertices, is there a Euler Path? What about a Circuit?

Yes path; No circuit

If we have 0 ODD vertices, is there a Euler Path? What about a Circuit?

Yes path; Yes ircuit