Math 118

Cheapest link

1. Choose the edge with the smallest weight. Mark this edge. 2. Choose the edge with the second smallest weight. Mark this edge. 3. Continue choosing and marking edges in order of increasing weight. (We must avoid creating a vertex of degree 3. We must avoid creating the circuit before we visit every vertex.)

Brute Force Algorithm

1. List all the Hamilton circuits in a graph. 2. Compute all of their weights. 3. Pick the one with the lowest weight.

unicursal tracing property

A graph has the ________________ if you can draw it without picking up your pen or retracing.

connected

A graph is ________ if you can reach any vertex by traversing the edges given in the graph.

completely defined; equivalent

A graph is __________ by its edge set and its vertex sets. This means two graphs that look very different can still be _________.

vertex; edge

A graph is a structure consisting of a set of objects (called the _____ set) and a list describing how the objects are related (called the ______ set).

exactly; bridge; vertices; n - 1

A network is a tree if: 1. If we pick any two vertices in the network, there is ______ one path between them. 2. Every edge of the network is a ______. 3. If there are n _____, there are ________ edges.

tree

A network with no circuits

bridge

A path can only cross a ______ once.

edge

A path that travels over every _____ is called a Eulerian path.

Eulerian

A path that travels over every edge is called an ________ path.

exhaustive route

A route that travels over every edge in the graph AT LEAST once

Hamilton circuit

A tour that is a circuit (same start and end) is called a ____________.

Hamilton path

A tour that is a path is called a __________.

network; circuits

A tree is a _______ with no _______.

adjacent edges; edge; once

A trip is a series of ___________ with the property that an ______ can only be traveled ______.

tour

A trip that repeats no edges and visits every vertex in the graph exactly once.

edge

An Euler circuit is a circuit that traverses every ______ exactly once.

optimal Eulerization

An Eulerization that adds as few duplicate edges as possible is called an ___________.

Euler path

An _______ in a graph is a path that traverses every edge exactly once.

Euler circuit

An ___________ is a circuit that traverses every edge exactly once.

Hamilton path

Any graph with a Hamilton circuit also has a _________.

Hamilton circuit

Any graph with a ______________ also has a Hamilton path.

component

Any graph with more than one __________ is not going to be connected.

Repetitive Nearest Neighbor

Apply nearest neighbor at every vertex, and pick the best one.

efficient

Basic objective: to construct specially designed graphs, to find the most _________ solutions to real-world problems

Hamilton circuits; weights; lowest weight

Brute force algorithm: 1. List all the ___________ in a graph. 2. Compute all of their ______. 3. Pick the one with the ________ _________.

smallest weight

Cheapest link: 1. Choose the edge with the _________. Mark this edge. 2. Choose the edge with the second __________. Mark this edge. 3. Continue choosing and marking edges in order of increasing weight.

vertices; edges

Do NOT repeat ______ or ______ when making hamilton paths/circuits.

vertex; 3; circuit

During Cheapest Link: 1. We must avoid creating a _____ of degree ___. We must avoid creating the _____ before we visit every vertex.

Relative error

E = (C - opt)/opt

path

Fleury's Algorithm to find Euler ______: Start at a vertex of odd degree.

circuit

Fleury's Algorithm to find Euler ________: Start at any vertex; At each step, pick the next edge in the path to not be a bridge of the part of the graph not yet traveled, if possible. If not, pick the bridge. Continue until done.

odd vertices

Optimal Eulerization requires adding at least half as many duplicate edges as there are _________.

0

R = ____ for any tree

weight; circuit; weight; optimal tour

Relative error equation: E = (C - opt)/opt, where C equals the ______ of the ______ that was found, and opt equals the ______ of the ____________.

organic

Some networks evolve without much planning. These are called ________ networks.

redundancy

Suppose we have a network with N vertices and M edges. Then the _________ of the network is R = M-(N-1) = M-N+1

R = M-(N-1) = M-N+1

Suppose we have a network with N vertices and M edges. Then the redundancy of the network is _________________.

N vertices; M edges

Suppose we have a network with ________ and _______. Then the redundancy of the network is R = M-(N-1) = M-N+1.

weight; weight; edges

The _______ of a Hamilton circuit or path is the sum of the ______s of its individual ______.

odd vertices

A graph always has an even number of _________.

Euler path

A graph has an open unicursal tracing if it has an ___________.

unicursal tracing

A graph has a _________ if you can draw the entire graph without picking up your pen.

Euler circuit

A graph has a closed unicursal tracing if it has an _____________.

open

A graph has an _______ unicursal tracing if it has an Euler path.

Nearest Neighbor Algorithm

1. Start at the designated starting vertex, or any vertex if there isn't one. 2. From the starting vertex, go to its nearest neighbor (vertex for which corresponding edge has the lowest weight). 3. From the next vertex, go to the nearest neighbor that has not yet been visited. 4. Continue until done.

trip

A ____ is a sequence of adjacent edges with the property that an edge can only be traveled once.

path

A _____ has different start and end points.

clique

A ______ in a graph is a subset of vertices that are all interconnected.

graph

A ______ is a structure consisting of a set of objects (called the vertex set) and a list describing how the objects are related (called the edge set).

circuit

A _______ has the same start and end point.

network

A _______ is just a connected graph.

bridge

A _______ is the only way from one part of a graph to another. (A graph can have more than one of these.)

degree

A _______ of a vertex is the number of edges originating in that vertex.

complete graph

A ________ on n vertices (Kn) is a graph with every possible edge included.

closed

A ________ unicursal tracing has the same start and end point.

connected; 2

A _________ graph has an Euler path when it has exactly ____ odd vertices.

vertices; interconnected

A clique in a graph is a subset of _______ that are all ___________.

even

A graph always has an _______ number of odd vertices.

redundancy

How far is this network from being a tree? (How many edges would we have to remove from this network so that the resulting network is a tree?)

twice

How many times must you count a loop?

exactly two

If a connected graph has __________ vertices, then it does have an Euler path.

no odd vertices

If a connected graph has ____________, then it has an Euler circuit. Otherwise, it has no Euler circuit.

Euler Circuit Theorem

If a connected graph has no odd vertices, then it has an Euler circuit. Otherwise, it has no Euler circuits.

Euler circuit

If a connected graph has no odd vertices, then it has an _________. Otherwise, it has no ________.

vertices; exactly; path

If we pick any two ________ in a tree, there is ______ one _____ between them.

connected

If you have a graph that's not _______, there's no way you have Euler paths or circuits.

edge

If you have any Hamilton circuit and take away an ________, it becomes a Hamilton path.

Hamilton path

If you have any Hamilton circuit and take away an edge, it becomes a ___________.

edge

In a Hamilton circuit or path, we do NOT need to traverse every _____ of the graph.

edge

In a Hamilton circuit or path, we do not need to traverse every _____ of the graph.

n-1

In a complete graph: In Kn, the degree of every vertex is _____.

(n-1)!

In a complete graph: The number of Hamilton circuits in Kn is _______.

(n(n-1))/2

In a complete graph: The number of edges in Kn is _________.

more than two

In a connected graph with ________ odd vertices, there is no Euler path.

odd vertices

In a connected graph with more than two __________, there is no Euler path. If a connected graph has exactly two _________, then it does have an Euler path.

Euler Path Theorem

In a connected graph with more than two odd vertices, there is no Euler path. If a connected graph has exactly two odd vertices, then it does have an Euler path.

degree of separation

In a network with two vertices A and B, the ____________ between A and B is the length of the shortest path between them.

bridge

In a tree, every edge is a _______.

n - 1

In a tree, if there are n vertices, there are _______ edges.

nearest neighbor

Nearest Neighbor Algorithm: 1. Start at the designated starting vertex, or any vertex if there isn't one. 2. From the starting vertex, go to its __________ (vertex for which corresponding edge has lowest weigh). 3. Continue until done.

degrees; vertices

The sum of the _______ of all the _______ in a graph is always twice the number of edges in a graph.

twice; edges

The sum of the degrees of all the vertices in a graph is always ______ the number of _______ in a graph.

complete graph

There are special types of graphs for which we know hamilton paths and circuits always exist. These are called _________.

Fleury's Algorithm

To find Euler path: Start at vertex of odd degree; At each step, pick the next edge in the path to not be a bridge of the part of the graph not yet traveled, if possible. If not, pick the bridge. Continue until done.

loop

We can only have a circuit of length 1 if we have a _______.

multiedge

We can only have a circuit of length 2 if we have a _______.

Optimal exhaustive route

an exhaustive route that uses the fewest edges possible

E = (C-opt)/opt

relative error equation

Eulerizing

the process of adding duplicate edges to a graph so that the resulting graph has an Euler circuit is called _________ a graph

Semi-Eulerization

the process of adding duplicate edges to a graph so that the resulting graph has an Euler path