Difference between revisions of "Graph Theory"

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== Overview ==
 
 
 
Many problems are naturally formulated in terms of points and connections between them.  For example, a computer network has PCs connected by cables, an airline map has cities connected by routes, and a school has rooms connected by hallways.  A graph is a mathematical object which models such situations.
 
Many problems are naturally formulated in terms of points and connections between them.  For example, a computer network has PCs connected by cables, an airline map has cities connected by routes, and a school has rooms connected by hallways.  A graph is a mathematical object which models such situations.
  
A graph is a collection of vertices and edges.  An edge is a connection between two vertices (or nodes).  One can draw a graph by marking points for the vertices and drawing lines connecting them for the edges, but it must be borne in mind that the graph is defined independently of the representation.  For example, the following two drawings represent the same graph:
+
== Overview ==
 
+
[[Graph A]]
+
  
 +
A ''graph'' is a collection of vertices and edges.  An ''edge'' is a connection between two ''vertices'' (sometimes referred to as ''nodes'').  One can draw a graph by marking points for the vertices and drawing lines connecting them for the edges, but the graph is defined independently of the visual representation.  For example, the following two drawings represent the same graph:
  
[[Graph B]]
+
::[[File:Graph fig1a.svg|150px]] <span style="display:inline-block; width: 5em;"></span>[[File:Graph fig1b.svg|150px]]
  
 +
The precise way to represent this graph is to identify its set of vertices {A, B, C, D, E, F, G}, and its set of edges between these vertices {AB, AD, BD, CF, FG, GH, GE, HE}.
  
[[Graph C]]
+
To be precise, the graph above is an ''undirected graph''; the edge from x to y is the same as from y to x.
  
 
== Terminology ==
 
== Terminology ==
  
The precise way to represent this graph is to identify its set of vertices {A, B, C, D, E, F, G, H, I, J, K, L, M}, and its set of edges between these vertices {AG, AB, AC, LM, JM, JL, JK, ED, FD, HI, FE, AF, GE}.
+
A ''path'' from vertex x to y in a graph is a list of vertices, in which successive vertices are connected by edges in the graph.  For example, FGHE is path from F to E in the graph above. A ''simple path'' is a path with no vertex repeated.  For example, FGHEG is not a simple path.
  
A path from vertex x to y in a graph is a list of vertices, in which successive vertices are connected by edges in the graph.  For example, BAFEG is path from B to G in the graph above.  A simple path is a path with no vertex repeated.  For example, BAFEGAC is not a simple path.
+
A graph is ''connected'' if there is a path from every vertex to every other vertex in the graph.  Intuitively, if the vertices were physical objects and the edges were strings connecting them, a connected graph would stay in one piece if picked up by any vertex.  A graph which is not connected is made up of ''connected components''.  For example, the graph above has two connected components: {A, B, D} and {C, E, F, G, H}.
  
A graph is connected if there is a path from every vertex to every other vertex in the graphIntuitively, if the vertices were physical objects and the edges were strings connecting them, a connected graph would stay in one piece if picked up by any vertexA graph which is not connected is made up of connected components.  For example, the graph above has three connected components: {I, H}, {J, K, L, M} and {A, B, C, D, E, F, G}.
+
A ''cycle'' is a path, which is simple except that the first and last vertex are the same (a path from a point back to itself)For example, the path HEGH is a cycle in our exampleVertices must be listed in the order that they are traveled to make the path; any of the vertices may be listed first.  Thus, HEGH and EHGE are different ways to identify the same cycle.  For clarity, we list the start / end vertex twice: once at the start of the cycle and once at the end.
  
A cycle is a path, which is simple except that the first and last vertex are the same (a path from a point back to itself).  For example, the path AFEGA is a cycle in our example.  Vertices must be listed in the order that they are traveled to make the path; any of the vertices may be listed first.  Thus, FEGAF and GAFEG are different ways to identify the same cycle.  For clarity, we list the start / end vertex twice: once at the start of the cycle and once at the end.
+
We’ll denote the number of vertices in a given graph by V and the number of edges by E.  Note that E can range anywhere from V to V<sup>2</sup> (or V(V-1)/2 in an undirected graph).  Graphs with all edges present are called ''complete graphs''; graphs with relatively few edges present (say less than V log(V)) are called ''sparse''; graphs with relatively few edges missing are called ''dense.''
 
+
We’ll denote the number of vertices in a given graph by V, the number of edges by E.  Note that E can range anywhere from V to V2 (or 1/2 V(V-1) in an undirected graph).  Graphs will all edges present are called complete graphs; graphs with relatively few edges present (say less than V log(V)) are called sparse; graphs with relatively few edges missing are called dense.
+
  
 
== Trees ==
 
== Trees ==
  
A graph with no cycles is called a tree.  There is only one path between any two nodes in a tree.  A tree on N vertices contains exactly N-1 edges.  A spanning tree of a graph is a subgraph that contains all the vertices and forms a tree.  A group of disconnected trees is called a forest.
+
A graph with no cycles is called a ''tree''.  There is only one path between any two nodes in a tree.  A tree on N vertices contains exactly N-1 edges.  A ''spanning tree'' of a graph is a subgraph that contains all the vertices and forms a tree.  Minimal spanning trees can be found for weighted graphs (i.e. each edge has a cost or distance associated with it) in order to minimize the cost or distance across an entire network.  A group of disconnected trees is called a ''forest.''
  
Directed graphs are graphs which have a direction associated with each edge.  An edge xy in a directed graph can be used in a path that goes from x to y but not necessarily from y to x.  For example, a directed graph similar to our example graph is drawn below:
+
== Directed Graphs ==
  
 +
''Directed graphs'' are graphs which have a direction associated with each edge.  An edge xy in a directed graph can be used in a path that goes from x to y but not necessarily from y to x.  For example, a directed graph similar to our example graph is drawn below:
  
 +
[[File:Graph fig1a directed.svg|150px]]
  
There is only one directed path from D to F.  Note that there are two edges between H and I, one each direction, which essentially makes an undirected edge.  An undirected graph can be thought of as a directed graph with all edges occurring in pairs in this way.  A dag (directed acyclic graph) is a directed graph with no cycles.
+
This graph is defined as the set of vertices V = {A,B,C,D,E,F,G,H} and the set of edges {AB,AD,DA,DB,EG,GE,HG,HE,GF,CF,FC}.  There is one directed path from G to C (namely, GFC); however, there are no directed paths from C to G.  Note that a few of the edges have arrows on both ends, such as the edge between A and D.  These dual arrows indicate that there is an edge in each direction, which essentially makes an undirected edge.  An ''undirected graph'' can be thought of as a directed graph with all edges occurring in pairs in this way.  A directed graph with no cycles is called a ''dag'' (directed acyclic graph).
  
 
== Adjacency Matrices ==
 
== Adjacency Matrices ==
  
It is frequently convenient to represent a graph by a matrix, as shown in the second sample problem below.  If we consider vertex A as 1, B as 2, etc., then a “one” in M at row i and column j indicates that there is a path from vertex i to j. If we raise M to the pth power, the resulting matrix indicates which paths of length p exist in the graph.  In fact, the quantity Mp(i,j) is the number of paths.
+
It is frequently convenient to represent a graph by a matrix, as shown in the second sample problem below.  If we consider vertex A as 1, B as 2, etc., then a “one” in M at row ''i'' and column ''j'' indicates that there is an edge from vertex ''i'' to ''j''., whereas a "zero" indicates there is not an edge. If we raise M to the ''pth'' power, the resulting matrix indicates which paths of length ''p'' exist in the graph.  The value in $M^p(i,j)$ is the number of paths from vertex ''i'' to vertex ''j''.
 
+
== Traversability ==
+
 
+
There are many applications of graphs and associated algorithms that will not be included in this topic except for the Elementary Division.  A Hamiltonian circuit is finding a path by which every vertex can be visited once and only once.  An Euler circuit is finding a path by which every edge can be used once and only once.  Graphs are traversable if there exists an Euler circuit unlike the famous Seven Bridges of Königsberg:
+
 
+
 
+
 
+
Rules of traversability show that vertices can be either odd or even based on how many edges are incident on it.  A graph is traversable if there are 0 or 2 odd vertices.  If there are 2 odd vertices, then the path must begin with one of them and end with the other.  With no odd edges, the path can begin with any vertex in the graph.  Conflict theory and proof of the Four Color Theorem also use principles of graph theory.
+
  
 
== Sample Problems ==
 
== Sample Problems ==
  
=== Sample Problem 1 ===
+
=== Problem 1 ===
  
 
Find the number of different cycles contained in the directed graph with vertices {A, B, C, D, E} and edges {AB, BA, BC, CD, DC, DB, DE}.
 
Find the number of different cycles contained in the directed graph with vertices {A, B, C, D, E} and edges {AB, BA, BC, CD, DC, DB, DE}.
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The graph is as follows:
 
The graph is as follows:
  
 +
::[[File:graph sample1.svg|196px]]
  
 
+
By inspection, the cycles are: ABA, BCDB, and CDC.  Thus, there are 3 cycles in the graph.
By inspection, the cycles are: {A,B}, {B,C,D} and {C,D}.  Thus, there are 3 cycles in the graph.
+
<!--
 
+
 
=== Sample Problem 2 ===
 
=== Sample Problem 2 ===
  
 
Given the directed graph below, how many more paths of length 3 would exist if edges AE and EA were added?
 
Given the directed graph below, how many more paths of length 3 would exist if edges AE and EA were added?
 +
 +
[[File:graph_s1.png]]
  
 
'''Solution:'''
 
'''Solution:'''
  
 
By using adjacency matrices, the current one is
 
By using adjacency matrices, the current one is
+
 
 +
[[File:matrix1_s2.png]]
 +
 
 
After adding edges AE and EA, the matrices are:
 
After adding edges AE and EA, the matrices are:
 +
 +
[[File:matrix2_s2.png]]
  
 
There are 6 more paths of length 2 since 16 – 10 = 6.
 
There are 6 more paths of length 2 since 16 – 10 = 6.
 +
-->
  
=== Sample Problem 3 ===
+
=== Problem 2===
  
List all possible values of x (5 bits long) that solve the following equation.
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In the following directed graph, find the total number of different paths from vertex A to vertex C of length 2 or 4.
:(LSHIFT-1 (10110 XOR (RCIRC-3 x) AND 11011)) = 01100
+
 
 +
::[[File:graph sample3.svg|128px]]
  
 
'''Solution:'''
 
'''Solution:'''
Since x is a string 5 bits long, represent it by abcde.  (RCIRC-3 x) is cdeab which, when ANDed with 11011 gives cd0ab.  This is XORed to 10110 to yield Cd1Ab (the capital letter is the NOT of its lower case).
 
Now, (LSHIFT-1 Cd1Ab) = d1Ab0 which has a value of 01100, we must have d=0, A=1 (hence a=0), b=0.  Thus, the solution must be in the form 00*0*, where * is an “I-don’t-care”.  The four possible values of x are: 00000, 00001, 00100 and 00101.
 
  
=== Sample Problem 4 ===
+
Let matrix $M$ represent the graph.  Recall that the number of paths from vertex $i$ to vertex $j$ of length $p$ equals $M^p[i,j]$.  The values of $M$, $M^2$ and $M^4$ are:
 +
 
 +
[[File:matrix_s3.png]]
 +
 
 +
There is 1 path of length 2 from A to C (cell [1,3] in $M^2$)
 +
and 3 paths of length 4 (cell [1,3] in $M^4$).
  
Evaluate the following expression:
+
=== Problem 3 ===
: ((RCIRC-14 (LCIRC-23 01101)) | (LSHIFT-1 10011) & (RSHIFT-2 10111))
+
  
'''Solution:'''
+
Given the adjacency matrix, draw the directed graph.
The problem can be rewritten as
+
: A | B & C
+
The AND has higher precedence than the OR.
+
  
The evaluation of expression A can be done in a straightforward way: LCIRC-23 011101) has a value of 01011, and RCIRC-14 01011) has a value of 10110. However, if the value of 23 and 14 were much larger, a more clever strategy is to offset the left and right circulates. For example, (RCIRC-7 (LCIRC-3 x)) would have the same results as (RCIRC-4 x). So, ((RCIRC-14 (LCIRC-23 01101)) has the same value as (LCIRC-9 01101), which has the same value as (LCIRC-4 01101). This is an easy expression to evaluate.
+
[[File:matrix_s4.png]]
  
Expressions B and C are pretty easy to evaluate:
+
'''Solution:'''
:B = (LSHIFT-1 10011) = 00110
+
:C = (RSHIFT-2 10111) = 00101
+
  
The expression becomes
+
There must be exactly 4 vertices: V = {A, B, C, D}. There must be be exactly 7 edges: E = {AB, AD, BA, BD, CA, DB, DC}.  Here are two valid drawings of the graph:
: A | B & C = 10110 | 00110 & 00101 = 10110 | 00100 = 10110
+
 
 +
: [[File:Graph sample4soln-a.svg|128px]] <span style="display:inline-block; width: 5em;"></span>[[File:Graph sample4soln-d.svg|128px]]
  
 
== Video Resources ==
 
== Video Resources ==
 +
 +
===ACSL Videos===
  
 
The following YouTube videos show ACSL students and advisors working out some ACSL problems that have appeared in previous contests. Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.  
 
The following YouTube videos show ACSL students and advisors working out some ACSL problems that have appeared in previous contests. Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.  
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{|
 
{|
 
|-
 
|-
| <youtube width="300" height="180">https://youtu.be/IeMsD3harrE</youtube>
+
| <youtube width="300" height="180">https://youtu.be/H96Lo2q_FSk</youtube>
| [https://youtu.be/IeMsD3harrE ''Bit String Flicking (Intro)'' ('''CalculusNguyenify''')]
+
| [https://youtu.be/H96Lo2q_FSk ''ACSL - Graph Theory Worksheet 1'' ('''misterminich''')]
 
+
A great two-part tutorial on this ACSL category. Part 1 covers bitwise operations AND, OR, NOT, and XOR.
+
  
 +
Shows the solution to the ACSL problem: '''Draw the directed graph with vertices A, B, C, D, E and the directed edges AB, BC, AD, BC, DC, ED, and EA.''
 
|-
 
|-
| <youtube width="300" height="180">https://youtu.be/jbKw8oYJPs4</youtube>
+
| <youtube width="300" height="180">https://youtu.be/y1fdIb_oNG0</youtube>
| [https://youtu.be/jbKw8oYJPs4 ''Bit String Flicking Shifts and Circs'' ('''CalculusNguyenify''')]
+
| [https://youtu.be/y1fdIb_oNG0 ''ACSL Graph Theory Worksheet 3'' ('''misterminich''')]
 
+
Part 2 covers logical shifts and circulate operations.
+
 
+
  
 +
Shows the solution to an ACSL problem asking to find how many paths of a specific length there are in a specific directed graph.
 
|-
 
|-
| <youtube width="300" height="180">https://youtu.be/XNBcO25mgCw</youtube>
+
| <youtube width="300" height="180">https://youtu.be/vtTCRMGB0K8</youtube>
| [https://youtu.be/XNBcO25mgCw ''Bit String Flicking'' ('''Tangerine Code''')]
+
| [https://youtu.be/vtTCRMGB0K8 ''Graph Theory ACSL Example Problem'' ('''Tangerine Code''')]
  
Shows the solution to the problem: (RSHIFT-3 (LCIRC-2 (NOT 10110)))
+
Shows the solution to an ACSL problem asking to find how many different cycles there are from a specific vertex in a specific directed graph.
 +
|}
  
 +
{|
 
|-
 
|-
| <youtube width="300" height="180">https://youtu.be/8J9AdxU5CW8</youtube>
+
| <youtube width="300" height="180">https://youtu.be/ivhOH6crQ1w</youtube>
| [https://youtu.be/8J9AdxU5CW8 ''Bit String Flicking by Ravi Yeluru'' ('''hemsra''')]
+
| [https://youtu.be/ivhOH6crQ1w ''ACSL Test Prep - Graph Theory'' ('''Mrs. Goopta''')]
 
+
Walks through two problems from the Junior Division.
+
 
+
|-
+
| <youtube width="300" height="180">https://youtu.be/aa_lQ8gft60</youtube>
+
| [https://youtu.be/aa_lQ8gft60 ''ACSL BitString Flicking Contest 2 Worksheet 1'' ('''misterminich''')]
+
 
+
Solves a handful of problems given in previous years at the Intermediate Division level.
+
  
 +
A talked-over presentation discussing graph theory as needed for the American Computer Science League and its tests.
 
|}
 
|}
  
 +
===Other Videos===
  
 +
There are dozens of YouTube videos that provide an introduction to graph theory. The following are a couple that we found particularly well done.  Be aware that the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.
  
<!--
 
 
{|
 
{|
 
|-
 
|-
| <youtube width="300" height="180">URL</youtube>
+
| <youtube width="300" height="180">https://youtu.be/44sQJK4BycY</youtube>
| [URL ''TITLE'' ('''AUTHOR''')]
+
| [https://youtu.be/44sQJK4BycY ''CS 106B Lecture: Graphs: basic concepts'']
  
DESCRIPTION
+
This lecture from Stanford University is s wonderful introduction to graph theory. The lecture starts out with many examples of real world problems that are modeled by graphs, and then moves on to review basic terminology relating to graph theory.
 +
|-
 +
| <youtube width="300" height="180"> https://youtu.be/gXgEDyodOJU</youtube>
 +
| [https://youtu.be/gXgEDyodOJU ''Data structures: Introduction to Graphs'' '''(mycodeschool)''']
 +
 
 +
A nice introduction to the fundamental concepts of graphs.
 +
 
|}
 
|}
-->
 

Latest revision as of 23:46, 9 June 2020

Many problems are naturally formulated in terms of points and connections between them. For example, a computer network has PCs connected by cables, an airline map has cities connected by routes, and a school has rooms connected by hallways. A graph is a mathematical object which models such situations.

Overview

A graph is a collection of vertices and edges. An edge is a connection between two vertices (sometimes referred to as nodes). One can draw a graph by marking points for the vertices and drawing lines connecting them for the edges, but the graph is defined independently of the visual representation. For example, the following two drawings represent the same graph:

Graph fig1a.svg Graph fig1b.svg

The precise way to represent this graph is to identify its set of vertices {A, B, C, D, E, F, G}, and its set of edges between these vertices {AB, AD, BD, CF, FG, GH, GE, HE}.

To be precise, the graph above is an undirected graph; the edge from x to y is the same as from y to x.

Terminology

A path from vertex x to y in a graph is a list of vertices, in which successive vertices are connected by edges in the graph. For example, FGHE is path from F to E in the graph above. A simple path is a path with no vertex repeated. For example, FGHEG is not a simple path.

A graph is connected if there is a path from every vertex to every other vertex in the graph. Intuitively, if the vertices were physical objects and the edges were strings connecting them, a connected graph would stay in one piece if picked up by any vertex. A graph which is not connected is made up of connected components. For example, the graph above has two connected components: {A, B, D} and {C, E, F, G, H}.

A cycle is a path, which is simple except that the first and last vertex are the same (a path from a point back to itself). For example, the path HEGH is a cycle in our example. Vertices must be listed in the order that they are traveled to make the path; any of the vertices may be listed first. Thus, HEGH and EHGE are different ways to identify the same cycle. For clarity, we list the start / end vertex twice: once at the start of the cycle and once at the end.

We’ll denote the number of vertices in a given graph by V and the number of edges by E. Note that E can range anywhere from V to V2 (or V(V-1)/2 in an undirected graph). Graphs with all edges present are called complete graphs; graphs with relatively few edges present (say less than V log(V)) are called sparse; graphs with relatively few edges missing are called dense.

Trees

A graph with no cycles is called a tree. There is only one path between any two nodes in a tree. A tree on N vertices contains exactly N-1 edges. A spanning tree of a graph is a subgraph that contains all the vertices and forms a tree. Minimal spanning trees can be found for weighted graphs (i.e. each edge has a cost or distance associated with it) in order to minimize the cost or distance across an entire network. A group of disconnected trees is called a forest.

Directed Graphs

Directed graphs are graphs which have a direction associated with each edge. An edge xy in a directed graph can be used in a path that goes from x to y but not necessarily from y to x. For example, a directed graph similar to our example graph is drawn below:

Graph fig1a directed.svg

This graph is defined as the set of vertices V = {A,B,C,D,E,F,G,H} and the set of edges {AB,AD,DA,DB,EG,GE,HG,HE,GF,CF,FC}. There is one directed path from G to C (namely, GFC); however, there are no directed paths from C to G. Note that a few of the edges have arrows on both ends, such as the edge between A and D. These dual arrows indicate that there is an edge in each direction, which essentially makes an undirected edge. An undirected graph can be thought of as a directed graph with all edges occurring in pairs in this way. A directed graph with no cycles is called a dag (directed acyclic graph).

Adjacency Matrices

It is frequently convenient to represent a graph by a matrix, as shown in the second sample problem below. If we consider vertex A as 1, B as 2, etc., then a “one” in M at row i and column j indicates that there is an edge from vertex i to j., whereas a "zero" indicates there is not an edge. If we raise M to the pth power, the resulting matrix indicates which paths of length p exist in the graph. The value in $M^p(i,j)$ is the number of paths from vertex i to vertex j.

Sample Problems

Problem 1

Find the number of different cycles contained in the directed graph with vertices {A, B, C, D, E} and edges {AB, BA, BC, CD, DC, DB, DE}.

Solution: The graph is as follows:

Graph sample1.svg

By inspection, the cycles are: ABA, BCDB, and CDC. Thus, there are 3 cycles in the graph.

Problem 2

In the following directed graph, find the total number of different paths from vertex A to vertex C of length 2 or 4.

Graph sample3.svg

Solution:

Let matrix $M$ represent the graph. Recall that the number of paths from vertex $i$ to vertex $j$ of length $p$ equals $M^p[i,j]$. The values of $M$, $M^2$ and $M^4$ are:

Matrix s3.png

There is 1 path of length 2 from A to C (cell [1,3] in $M^2$) and 3 paths of length 4 (cell [1,3] in $M^4$).

Problem 3

Given the adjacency matrix, draw the directed graph.

Matrix s4.png

Solution:

There must be exactly 4 vertices: V = {A, B, C, D}. There must be be exactly 7 edges: E = {AB, AD, BA, BD, CA, DB, DC}. Here are two valid drawings of the graph:

Graph sample4soln-a.svg Graph sample4soln-d.svg

Video Resources

ACSL Videos

The following YouTube videos show ACSL students and advisors working out some ACSL problems that have appeared in previous contests. Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.

ACSL - Graph Theory Worksheet 1 (misterminich)

Shows the solution to the ACSL problem: 'Draw the directed graph with vertices A, B, C, D, E and the directed edges AB, BC, AD, BC, DC, ED, and EA.

ACSL Graph Theory Worksheet 3 (misterminich)

Shows the solution to an ACSL problem asking to find how many paths of a specific length there are in a specific directed graph.

Graph Theory ACSL Example Problem (Tangerine Code)

Shows the solution to an ACSL problem asking to find how many different cycles there are from a specific vertex in a specific directed graph.

ACSL Test Prep - Graph Theory (Mrs. Goopta)

A talked-over presentation discussing graph theory as needed for the American Computer Science League and its tests.

Other Videos

There are dozens of YouTube videos that provide an introduction to graph theory. The following are a couple that we found particularly well done. Be aware that the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.

CS 106B Lecture: Graphs: basic concepts

This lecture from Stanford University is s wonderful introduction to graph theory. The lecture starts out with many examples of real world problems that are modeled by graphs, and then moves on to review basic terminology relating to graph theory.

Data structures: Introduction to Graphs (mycodeschool)

A nice introduction to the fundamental concepts of graphs.