GraphTheoryBasics

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Week 7 – Simple Graph

Algorithms

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A Few Graph Theory Applications

• _{Software tools (code flow, etc…)}

– _{McCabe IQ tool}

– Visual Paradigm

• Networks

– Flow, paths, etc…

• Game AI

– _{Path algorithms}

– Decision algorithms

• _Origami

– Planar models of 3d figures

• _{Modeling in general}

– Decision trees, Bayesian networks, neural networks

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What is a Graph

• A graph is a finite set of vertices that can be connected by undirected/directed

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Vertex

• _Vertex

– _{Represented as a circle} – _{Can also be called a node}

– _{Usually has some representational meaning}

(5)

Edge

• Edge

– _{Represented as a line} – _{Connects two vertices}

– Can have direction and weight

• Arc = directed edge

– Shown in red so arrowheads are visible

5

(6)

Face

• _{An area enclosed by edges and vertices} • Only for planar graphs

• Don’t forget the outside face

6

1

2

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How Many Vertices, Edges and Faces?

7

• _{V = ?} • F = ?

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How Many Vertices, Edges and Faces?

8

• _{V = 5} • F = 2

• E = 5

(9)

9

• _{V = ?} • F = ?

(10)

10

• _{V = 6} • F = 5

• E = 9

(11)

11

• _{V = ?} • F = ?

(12)

12

• _{V = 4} • F = 2

• E = 4

(13)

13

• _{V = ?} • F = ?

(14)

How Many Vertices, Edges and Faces?

14

• _{V = 4} • F = 4

• E = 6

(15)

Relationship between V, F, and E

15

What is the pattern/relationship?

V F E

Graph 1 5 2 5

Graph 2 6 5 9

Graph 3 4 2 4

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Euler's Formula

• _{For planar graphs} • V + F = E + 2

• V + F – E = 2

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Planar Graph

• _{A graph that can be drawn with no edges}

crossing each other

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Is This Graph Planar?

(19)

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Is This Graph Planar?

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Algorithm to determine planarity

• _{Auslander and Parter (1961; Skiena 1990, p. 247).} • _O(n^3)

(30)

Untangle

• _{Fun game}

• Try to draw the graph as a planar graph

• http://www.addictinggames.com/puzzle-games

/untangle.jsp

(31)

Sparseness

• _{Planar graphs are sparse}

– _{Most vertices do not share an edge}

• _{Matrix is mostly empty}

– _{Mostly 0’s}

(32)

Origami

• _{Planar graphs of 3D figures} • Map out graph on 2D surface

• Create 3D figure

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Example 1 - Hexahedron

(34)

Example 2 – Bucky ball (Hexagons = 0)

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Example 3 – Bucky ball (Hexagons = 20)

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Undirected Graph

• _{A graph that does not have any directed edges} • Implied that vertices have no self-edges

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Directed Graph

• _{A graph that has at least one directed edge}

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(38)

Degree

• _{The degree of a vertex is the number of edges}

it has

• The degree of a graph is the same as the maximum degree of all the vertices

• In a regular graph all vertices have the same degree

– _{Degree of a graph often refers to regular graphs}

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What is the Degree of Vertex A?

39

(40)

What is the Degree of this Vertex?

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A

(41)

What is the degree of this graph?

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(42)

Is this Graph Regular?

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No

(44)

Is This Graph Regular?

(45)

Is This Graph Regular?

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(46)

(47)

Path

• _{A path is traveling on the edges of the graph}

without visiting the same vertex more than once

• First vertex on the path can be the same as the last

• _{Paths do not have to end where they started}

(48)

Path Example

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A B C D

E F G H

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Is There a Path From A to L?

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A B C D

E F G H

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Is There a Path From A to L?

50

Yes

A B C D

E F G H

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Is There a Path From A to H?

51

A B C D

E F G H

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Is There a Path From A to H?

52

No

A B C D

E F G H

(53)

Path Example

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A B C D

E F G H

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Is There a Path From L to A

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A B C D

E F G H

(55)

Is There a Path From L to A

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No

A B C D

E F G H

(56)

Is There a Path From A to L

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A B C D

E F G H

(57)

Is There a Path From A to L

57

Yes

A B C D

E F G H

(58)

Trail

• _{A trail is traveling the edges of a graph}

without using the same edge more than once

• Can use a vertex more than once

• _{Trails do not have to end at the same vertex it}

starts at

(59)

Trail Example

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A B C D

E F G H

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Find Some Trails Starting at A and Ending at

D

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A B C D

E F G H

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Find Some Trails Starting at A and Ending at

D

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A  B  G  D

A  B  G  K  L  G  D

A  B  G  J  I  E  F  C  H  G  D

A B C D

E F G H

(62)

Cycle

• _{A cycle is a path that starts and ends at the same} vertex

• _A B  C  A • _A__C__B__A • _B__A__C__B • _B C  A  B • _C A  B  C • _C__B__A__C

(63)

Directed Cycle Example

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• A  B  C  A

• B  C  A  B

• C  A  B  C

A

B

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Circuit

• _{A circuit is a trail that starts and ends at the}

same vertex

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Find Some Circuits Starting at G

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Find Some Circuits Starting at G

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• _G__K__L__G

• _G J  I  E  F  C  H  G • _G C  H  G

A B C D

E F G H

(67)

Connected

• _{A graph is connected if there exists an}

undirected path from every vertex to every other vertex

(68)

Is This Graph Connected

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A B C D

E F G H

(69)

Is This Graph Connected?

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No

A B C D

E F G H

(70)

Tree

• _{A tree is a connected graph with no cycles}

(71)

Tree

71

A B C D

E F G H

(72)

(73)

Complete

• _{A complete graph is an undirected graph}

where every vertex has an edge to every other vertex

• _{All edges possible are in the graph}

(74)

Is This Graph Complete?

(75)

Is This Graph Complete?

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Hamilton Cycle

• _{A cycle that starts and ends at the same vertex}

and visits all other vertices exactly once

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Hamilton Cycle Example

• _{Red + Blue lines together = Hamilton cycle}

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Hamilton Cycle Example 2

• Red + Green lines together = Hamilton cycle

(83)

Eulerian Circuit (cycle)

• _{A circuit that starts and ends at the same}

vertex and uses every edge in the graph exactly once

• _http://

mathworld.wolfram.com/EulerianCycle.html

(84)

Does There Exists an Eulerian Path?

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A B C D

E F G H

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Does There Exists an Eulerian Path?

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• An Eulerian path exists if

• Undirected

– _{All vertices have even degree}

• Directed

– _{All vertices have the same in-degree as out-degree} – _{Exactly two vertices have odd degree}

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A B C D

E F G H

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A B C D

E F G H

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Does There Exists an Eulerian Path?

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A B C D

E F G H

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A B C D

E F G H

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90

A B C D

E F G H

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91

A B C D

E F G H

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92

A B C D

E F G H

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93

A B C D

E F G H

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94

A B C D

E F G H

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A B C D

E F G H

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Does There Exists an Eulerian Path?

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A B C D

E F G H

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A B C D

E F G H

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98

A B C D

E F G H

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A B C D

E F G H

(100)

100

A B C D

E F G H

(101)

101

A B C D

E F G H

(102)

Does There Exists an Eulerian Circuit?

102

• An Eulerian circuit exists if

• Undirected

– _{All vertices have even degree}

• Directed

(103)

How to Represent Graphs

• _{Multidimensional arrays / Adjacency matrix}

– _{Better for dense graphs}

• _{Dense = lots of edges}

• Edge list

– _{Better for sparse graphs}

• Sparse = few edges

(104)

Adjacency Matrix

• _{An N X N matrix where N is the number of}

vertices

• A ‘0’ represents to edge between the vertices

• _{A ‘1’ represents an edge between the vertices}

(105)

Undirected Adjacency Matrix

105

0 1 2 3 0 0 0 1 1

1 0 0 0 1

2 1 0 0 1

3 1 1 1 0

0

2 3

(106)

Directed Adjacency Matrix

106

0

2 3

1

0 1 2 3 0 0 0 1 1

1 0 0 0 1

2 0 0 0 1

(107)

Algorithms Using Adjacency Matrices

• _{Finding a path between two vertices} • Finding a trail between two vertices

– _{Same as path for undirected graphs} – _{May be different for directed graphs}

• _{Does the graph have a cycle} • Does the graph have a circuit

(108)

Algorithms Using Adjacency Matrices

• _{Is the graph connected} • Is the graph a tree

• Is the graph complete

• Does a graph have a Hamilton cycle – _{And what is the cycle}

• Does a graph have an Eulerian circuit – _{And what is the circuit}

http://www.addictinggames.com/puzzle-games/untangle.jsp

http://mathworld.wolfram.com/EulerianCycle.html