Podcast
Questions and Answers
What is a planar graph?
What is a planar graph?
- A graph where edges can cross each other
- A graph that can be drawn without edges crossing (correct)
- A graph with no vertices
- A graph that includes a Hamiltonian circuit
A graph is considered planar if at least one of its edges crosses another.
A graph is considered planar if at least one of its edges crosses another.
False (B)
What term is used to describe a graph whose edges and vertices come from a given graph that is not planar?
What term is used to describe a graph whose edges and vertices come from a given graph that is not planar?
subgraph
If a graph can be redrawn without any __________ between edges, it is classified as a planar graph.
If a graph can be redrawn without any __________ between edges, it is classified as a planar graph.
Match the following terms with their definitions:
Match the following terms with their definitions:
Which of the following statements about the Utilities Graph is true?
Which of the following statements about the Utilities Graph is true?
Redrawing edges such that they do not cross is always possible for any graph.
Redrawing edges such that they do not cross is always possible for any graph.
What main problem arises when trying to create a planar drawing of the Utilities Graph?
What main problem arises when trying to create a planar drawing of the Utilities Graph?
Which of the following statements is true about an Euler circuit?
Which of the following statements is true about an Euler circuit?
An Euler circuit can exist in a graph if at least one vertex has an odd degree.
An Euler circuit can exist in a graph if at least one vertex has an odd degree.
What is the term used for the number of edges that meet at a vertex?
What is the term used for the number of edges that meet at a vertex?
A circuit that uses every edge exactly once is called an ______ circuit.
A circuit that uses every edge exactly once is called an ______ circuit.
In order for a graph to be Eulerian, which condition must be satisfied?
In order for a graph to be Eulerian, which condition must be satisfied?
The path A–B–C–E–H–G–E–B–D–A is an Euler circuit.
The path A–B–C–E–H–G–E–B–D–A is an Euler circuit.
How can Euler prove that a graph cannot have an Euler circuit?
How can Euler prove that a graph cannot have an Euler circuit?
What time will it be 8 hours after 9 o’clock on a 12-hour clock?
What time will it be 8 hours after 9 o’clock on a 12-hour clock?
What is the minimum number of colors required to color the graph represented by the fictional map according to the content?
What is the minimum number of colors required to color the graph represented by the fictional map according to the content?
A graph can be colored using only two colors if it has a ring of five vertices.
A graph can be colored using only two colors if it has a ring of five vertices.
If the time now is 10 o'clock, then 7 hours ago it was 3 o'clock.
If the time now is 10 o'clock, then 7 hours ago it was 3 o'clock.
What day of the week is 16 days after Monday?
What day of the week is 16 days after Monday?
What is the term used to describe a graph that can be colored with four colors?
What is the term used to describe a graph that can be colored with four colors?
If today is Friday, then 14 days from now will also be _____
If today is Friday, then 14 days from now will also be _____
The four-color theorem states that every planar graph can be colored with ______ colors.
The four-color theorem states that every planar graph can be colored with ______ colors.
Match the following terms related to graph coloring with their definitions:
Match the following terms related to graph coloring with their definitions:
Match the day with its corresponding number in modular arithmetic:
Match the day with its corresponding number in modular arithmetic:
Is the congruence 29 ≡ 8 mod 3 true or false?
Is the congruence 29 ≡ 8 mod 3 true or false?
In the fictional map example, how is connectivity between neighboring countries represented in the graph?
In the fictional map example, how is connectivity between neighboring countries represented in the graph?
According to the four-color theorem, it is possible for a planar graph to require more than four colors.
According to the four-color theorem, it is possible for a planar graph to require more than four colors.
The remainder of 16 divided by 7 is 2.
The remainder of 16 divided by 7 is 2.
What strategy can be used to color the graph vertices effectively?
What strategy can be used to color the graph vertices effectively?
What was the day of the week on July 4, 2017?
What was the day of the week on July 4, 2017?
What day of the week is July 4, 2022?
What day of the week is July 4, 2022?
The year 2020 is not a leap year.
The year 2020 is not a leap year.
What is the result of $61 \mod 12$?
What is the result of $61 \mod 12$?
To find what time it was 57 hours ago from 5 o’clock, calculate $(5 - 57) \mod ______$.
To find what time it was 57 hours ago from 5 o’clock, calculate $(5 - 57) \mod ______$.
Match the operations with their results:
Match the operations with their results:
What does the expression 1826 modulo 7 represent?
What does the expression 1826 modulo 7 represent?
In modular arithmetic, the result of an operation mod n is always less than n.
In modular arithmetic, the result of an operation mod n is always less than n.
How many extra days are counted in a leap year?
How many extra days are counted in a leap year?
What is the result of the expression (15 * 23) mod 11?
What is the result of the expression (15 * 23) mod 11?
The check digit of an ISBN is calculated using a multiplication modulo system.
The check digit of an ISBN is calculated using a multiplication modulo system.
How many digits are there in a standard ISBN number?
How many digits are there in a standard ISBN number?
If an integer x is 5, then the expression (x - 57) mod 12 results in __.
If an integer x is 5, then the expression (x - 57) mod 12 results in __.
Match the following ISBN components with their descriptions:
Match the following ISBN components with their descriptions:
What is the purpose of the check digit in an ISBN?
What is the purpose of the check digit in an ISBN?
Transposition errors are rarely caught by the ISBN coding system.
Transposition errors are rarely caught by the ISBN coding system.
What happens when an incorrect ISBN is sent?
What happens when an incorrect ISBN is sent?
What does each dot in a graph typically represent in social network connections?
What does each dot in a graph typically represent in social network connections?
In a graph, all edges must begin and end at vertices.
In a graph, all edges must begin and end at vertices.
Who is the student that has only one study group in common with others?
Who is the student that has only one study group in common with others?
Amber is involved in __________ study groups.
Amber is involved in __________ study groups.
Match the following students with their number of study groups:
Match the following students with their number of study groups:
Which type of connection in a graph indicates that two elements share a common attribute?
Which type of connection in a graph indicates that two elements share a common attribute?
Multiple edges can exist between the same two vertices.
Multiple edges can exist between the same two vertices.
How many study groups does Laura have in common with the others?
How many study groups does Laura have in common with the others?
What is the purpose of a check digit in ISBN and UPC codes?
What is the purpose of a check digit in ISBN and UPC codes?
The Luhn algorithm is used to validate credit card numbers based on modular arithmetic.
The Luhn algorithm is used to validate credit card numbers based on modular arithmetic.
What is the final sum required for the Luhn algorithm to determine a valid credit card number?
What is the final sum required for the Luhn algorithm to determine a valid credit card number?
The check digit for the ISBN of 'The Equation that Couldn't Be Solved' is _____
The check digit for the ISBN of 'The Equation that Couldn't Be Solved' is _____
Match the following coding systems with their descriptions:
Match the following coding systems with their descriptions:
Which of the following statements about UPC is NOT true?
Which of the following statements about UPC is NOT true?
UPC codes and ISBNs use the same method to calculate their check digits.
UPC codes and ISBNs use the same method to calculate their check digits.
What is the typical length of a credit card number?
What is the typical length of a credit card number?
Which statement about an Euler circuit is true?
Which statement about an Euler circuit is true?
For a graph to be Eulerian, at least one vertex must have an odd degree.
For a graph to be Eulerian, at least one vertex must have an odd degree.
How can one determine if a graph has an Euler circuit?
How can one determine if a graph has an Euler circuit?
A path that begins and ends at the same vertex is known as a ______.
A path that begins and ends at the same vertex is known as a ______.
Which of the following is an example of an Euler circuit?
Which of the following is an example of an Euler circuit?
An Euler circuit can exist in a graph if it has at least one vertex of even degree.
An Euler circuit can exist in a graph if it has at least one vertex of even degree.
What observation did Euler make regarding the degree of vertices for an Euler circuit to exist?
What observation did Euler make regarding the degree of vertices for an Euler circuit to exist?
What is the result of calculating $15 mod 6$?
What is the result of calculating $15 mod 6$?
If today is Thursday, then 6 days from now it will be Thursday.
If today is Thursday, then 6 days from now it will be Thursday.
How many hours ago was it if the current time is 2 o’clock and it was at 9 o’clock?
How many hours ago was it if the current time is 2 o’clock and it was at 9 o’clock?
The result of $8 + 5$ on a 12-hour clock is _____
The result of $8 + 5$ on a 12-hour clock is _____
Match the days of the week with their corresponding numbers in modular arithmetic:
Match the days of the week with their corresponding numbers in modular arithmetic:
What day of the week is represented by the number 2 in modular arithmetic?
What day of the week is represented by the number 2 in modular arithmetic?
The congruence statement $29  8$ mod 3 is true.
The congruence statement $29  8$ mod 3 is true.
What will be the time 7 hours ago if it is currently 9 o'clock?
What will be the time 7 hours ago if it is currently 9 o'clock?
What is the minimum number of colors required to color the fictional map described in the content?
What is the minimum number of colors required to color the fictional map described in the content?
The graph representing the fictional map is always 3-colorable.
The graph representing the fictional map is always 3-colorable.
What technique is suggested for coloring the vertices of a graph efficiently?
What technique is suggested for coloring the vertices of a graph efficiently?
Match the following concepts with their descriptions:
Match the following concepts with their descriptions:
How is the connectivity between neighboring countries represented in the graph?
How is the connectivity between neighboring countries represented in the graph?
A graph can be colored with only three colors if it has a ring of five vertices.
A graph can be colored with only three colors if it has a ring of five vertices.
The vertices of the graph are colored to ensure that no two vertices connected by an edge share the same ______.
The vertices of the graph are colored to ensure that no two vertices connected by an edge share the same ______.
What is the relationship described by Euler's formula for a planar graph?
What is the relationship described by Euler's formula for a planar graph?
In graph theory, the region surrounding a graph is called the infinite face.
In graph theory, the region surrounding a graph is called the infinite face.
How many faces are counted in a planar graph with seven edges and five vertices, according to Euler's formula?
How many faces are counted in a planar graph with seven edges and five vertices, according to Euler's formula?
In graph theory, which of the following is NOT a practical application of graph coloring?
In graph theory, which of the following is NOT a practical application of graph coloring?
What is the term used for a graph that can be drawn on a plane without edges overlapping?
What is the term used for a graph that can be drawn on a plane without edges overlapping?
According to the four-color theorem, a planar graph may require more than four colors for proper coloring.
According to the four-color theorem, a planar graph may require more than four colors for proper coloring.
What is the minimum number of colors required to color the graph represented by the fictional map?
What is the minimum number of colors required to color the graph represented by the fictional map?
The four-color theorem guarantees that a planar graph can use only three colors.
The four-color theorem guarantees that a planar graph can use only three colors.
How does the connectivity between neighboring countries get represented in a graph?
How does the connectivity between neighboring countries get represented in a graph?
The graph in the example requires a minimum of _____ colors for proper vertex coloring.
The graph in the example requires a minimum of _____ colors for proper vertex coloring.
What is the total weight of the Hamiltonian circuit A-D-B-F-C-E-A?
What is the total weight of the Hamiltonian circuit A-D-B-F-C-E-A?
The Hamiltonian circuit mentioned can only be traversed in one direction.
The Hamiltonian circuit mentioned can only be traversed in one direction.
What is the primary purpose of using weights in a graph?
What is the primary purpose of using weights in a graph?
The cost of traveling between cities can be represented as the weight of the _____ in a graph.
The cost of traveling between cities can be represented as the weight of the _____ in a graph.
Match the following cities with the corresponding edge weights in the context of travel costs:
Match the following cities with the corresponding edge weights in the context of travel costs:
Which algorithm is mentioned for finding a low-cost route visiting each city exactly once?
Which algorithm is mentioned for finding a low-cost route visiting each city exactly once?
The edge-picking algorithm guarantees an optimal route in all cases.
The edge-picking algorithm guarantees an optimal route in all cases.
What was the smallest weight edge traveled to from Paris in the example?
What was the smallest weight edge traveled to from Paris in the example?
The route A–D–B–F–C–E–A completes a _____ circuit.
The route A–D–B–F–C–E–A completes a _____ circuit.
What happens when marking edges during the algorithm?
What happens when marking edges during the algorithm?
What condition must be met for a graph to contain an Euler circuit?
What condition must be met for a graph to contain an Euler circuit?
An Euler path can exist in a graph with all vertices having even degrees.
An Euler path can exist in a graph with all vertices having even degrees.
In the context of the Euler paths, which two cities had to be the start and end points for the photographer's trip?
In the context of the Euler paths, which two cities had to be the start and end points for the photographer's trip?
Euler paths traverse every edge in a graph exactly ______.
Euler paths traverse every edge in a graph exactly ______.
Match the following terms related to Euler circuits and paths with their definitions:
Match the following terms related to Euler circuits and paths with their definitions:
Which of the following represents a valid Euler circuit for the given graph?
Which of the following represents a valid Euler circuit for the given graph?
There are multiple Euler circuits possible in an Eulerian graph.
There are multiple Euler circuits possible in an Eulerian graph.
In a complete graph with six vertices, what can be said about the circuits found by algorithms?
In a complete graph with six vertices, what can be said about the circuits found by algorithms?
The greedy algorithm selects the highest weight edge at each step.
The greedy algorithm selects the highest weight edge at each step.
What is the total weight of the Hamiltonian circuit found using the greedy algorithm starting at vertex A?
What is the total weight of the Hamiltonian circuit found using the greedy algorithm starting at vertex A?
The edge with the smallest weight in the example was __________.
The edge with the smallest weight in the example was __________.
Which edge was chosen from vertex E in the greedy algorithm example?
Which edge was chosen from vertex E in the greedy algorithm example?
Match the algorithms with their characteristics:
Match the algorithms with their characteristics:
In the edge-picking algorithm, edges with the same weight can be represented together.
In the edge-picking algorithm, edges with the same weight can be represented together.
What is Euler's formula related to in the context of planar graphs?
What is Euler's formula related to in the context of planar graphs?
What is a characteristic feature of a complete graph?
What is a characteristic feature of a complete graph?
In a planar graph, the exterior region is considered as one of the faces.
In a planar graph, the exterior region is considered as one of the faces.
What is the minimum number of colors required to color any planar graph according to the four-color theorem?
What is the minimum number of colors required to color any planar graph according to the four-color theorem?
What is the first edge selected when using the edge-picking algorithm in the weighted graph?
What is the first edge selected when using the edge-picking algorithm in the weighted graph?
In Euler's formula, if a graph has 5 vertices and 4 faces, then it must have ______ edges.
In Euler's formula, if a graph has 5 vertices and 4 faces, then it must have ______ edges.
Match the following terms with their definitions related to graph theory:
Match the following terms with their definitions related to graph theory:
What does the connection between map coloring and graph theory allow for?
What does the connection between map coloring and graph theory allow for?
A graph that can be colored with only two colors can contain cycles of any length.
A graph that can be colored with only two colors can contain cycles of any length.
What happens when neighboring countries share a border in graph theory?
What happens when neighboring countries share a border in graph theory?
The region surrounding the graph in a planar drawing is known as the ______ face.
The region surrounding the graph in a planar drawing is known as the ______ face.
According to the nonplanar graph theorem, which of the following graphs cannot be contracted to K5?
According to the nonplanar graph theorem, which of the following graphs cannot be contracted to K5?
What is required for a graph to be classified as Eulerian?
What is required for a graph to be classified as Eulerian?
An Euler path requires returning to the starting point.
An Euler path requires returning to the starting point.
Name a city with an odd degree in the road example provided.
Name a city with an odd degree in the road example provided.
A path that visits every vertex exactly once is called a ______ circuit.
A path that visits every vertex exactly once is called a ______ circuit.
How many vertices with odd degrees must exist for an Euler path to be possible?
How many vertices with odd degrees must exist for an Euler path to be possible?
All Euler circuits must cover each edge of the graph exactly twice.
All Euler circuits must cover each edge of the graph exactly twice.
What must a photographer do to successfully traverse all roads exactly once in the city map?
What must a photographer do to successfully traverse all roads exactly once in the city map?
An example of an odd degree vertex in the road map is ______.
An example of an odd degree vertex in the road map is ______.
What is a complete graph?
What is a complete graph?
A graph is considered connected if all vertices can be reached from any other vertex.
A graph is considered connected if all vertices can be reached from any other vertex.
What is the chromatic number of a planar graph according to the four-color theorem?
What is the chromatic number of a planar graph according to the four-color theorem?
What do you call an edge that begins and ends at the same vertex?
What do you call an edge that begins and ends at the same vertex?
A nonplanar graph always requires four or fewer colors to color it.
A nonplanar graph always requires four or fewer colors to color it.
A graph without any edges is considered __________.
A graph without any edges is considered __________.
What is the minimum number of colors needed to color a graph so that no two connected vertices have the same color?
What is the minimum number of colors needed to color a graph so that no two connected vertices have the same color?
Which of the following best describes equivalent graphs?
Which of the following best describes equivalent graphs?
In scheduling meetings among clubs with shared members, the number of required time slots corresponds to the graph's __________ for proper coloring.
In scheduling meetings among clubs with shared members, the number of required time slots corresponds to the graph's __________ for proper coloring.
What is the significance of the edges in a graph?
What is the significance of the edges in a graph?
Match the graph type with its continuous coloring conditions:
Match the graph type with its continuous coloring conditions:
What is the configuration of edges, vertices, and faces in a planar graph that verifies Euler’s formula?
What is the configuration of edges, vertices, and faces in a planar graph that verifies Euler’s formula?
In a planar graph, the infinite face is not considered a face.
In a planar graph, the infinite face is not considered a face.
What is Euler's formula for a planar graph?
What is Euler's formula for a planar graph?
A planar graph can be drawn in such a way that no edges __________.
A planar graph can be drawn in such a way that no edges __________.
Match each term with its corresponding definition:
Match each term with its corresponding definition:
How many colors are needed to color a map of countries, ensuring no two neighbors share the same color, as indicated by the content?
How many colors are needed to color a map of countries, ensuring no two neighbors share the same color, as indicated by the content?
Graph coloring can be applied in various practical applications, including scheduling tasks.
Graph coloring can be applied in various practical applications, including scheduling tasks.
When counting the faces in a planar graph, what additional face must always be included?
When counting the faces in a planar graph, what additional face must always be included?
In graph theory, an edge connects two __________.
In graph theory, an edge connects two __________.
A country is represented by a vertex in graph theory if it __________.
A country is represented by a vertex in graph theory if it __________.
The graph in Figure 5.23 is 4-colorable.
The graph in Figure 5.23 is 4-colorable.
A map can be colored using the fewest colors possible with __________ where no two adjacent countries share the same color.
A map can be colored using the fewest colors possible with __________ where no two adjacent countries share the same color.
Match the following graph-coloring terms with their descriptions:
Match the following graph-coloring terms with their descriptions:
Which is true about the four-color theorem?
Which is true about the four-color theorem?
To color a graph effectively, you can reuse colors when connecting non-adjacent vertices.
To color a graph effectively, you can reuse colors when connecting non-adjacent vertices.
If a graph requires fewer colors than the maximum, it is referred to as being __________.
If a graph requires fewer colors than the maximum, it is referred to as being __________.
Why will a particular vertex need a fourth color in the example provided?
Why will a particular vertex need a fourth color in the example provided?
What is the remainder when the product (15 * 23) is divided by 11?
What is the remainder when the product (15 * 23) is divided by 11?
The check digit of an ISBN is calculated using an addition modulo system.
The check digit of an ISBN is calculated using an addition modulo system.
If the current time is 5 o'clock, what time was it 57 hours ago?
If the current time is 5 o'clock, what time was it 57 hours ago?
The first three digits of an ISBN are always _____ or 979.
The first three digits of an ISBN are always _____ or 979.
Match the following components of an ISBN with their descriptions:
Match the following components of an ISBN with their descriptions:
Which of the following statements correctly describes the traveling salesman problem?
Which of the following statements correctly describes the traveling salesman problem?
The Hamiltonian circuit must visit each city exactly once.
The Hamiltonian circuit must visit each city exactly once.
What is the total distance travelled by the route Chicago–New York–Dallas–Philadelphia–Atlanta–Washington, D.C.–Chicago?
What is the total distance travelled by the route Chicago–New York–Dallas–Philadelphia–Atlanta–Washington, D.C.–Chicago?
The two algorithms mentioned for finding a solution in complete graphs are the greedy algorithm and the __________ algorithm.
The two algorithms mentioned for finding a solution in complete graphs are the greedy algorithm and the __________ algorithm.
Match the following routes with their total distances:
Match the following routes with their total distances:
Which statement about complete graphs is true?
Which statement about complete graphs is true?
Finding the optimal Hamiltonian circuit in a weighted graph has a known shortcut.
Finding the optimal Hamiltonian circuit in a weighted graph has a known shortcut.
What is the total distance for the route Chicago–Washington, D.C.–Dallas–New York–Atlanta–Philadelphia–Chicago?
What is the total distance for the route Chicago–Washington, D.C.–Dallas–New York–Atlanta–Philadelphia–Chicago?
In the context of the traveling salesman problem, a route that visits each city only once is termed a __________ circuit.
In the context of the traveling salesman problem, a route that visits each city only once is termed a __________ circuit.
Which of the following statements accurately describes a Hamiltonian circuit?
Which of the following statements accurately describes a Hamiltonian circuit?
Dirac’s Theorem guarantees that a graph is Hamiltonian if every vertex has a degree of at least $n/2$.
Dirac’s Theorem guarantees that a graph is Hamiltonian if every vertex has a degree of at least $n/2$.
What does a weighted graph associate with each edge?
What does a weighted graph associate with each edge?
A sequence of flights that starts and ends at the same city without visiting any city twice is represented by a Hamiltonian __________.
A sequence of flights that starts and ends at the same city without visiting any city twice is represented by a Hamiltonian __________.
Match the following theorems or concepts with their correct descriptions:
Match the following theorems or concepts with their correct descriptions:
What is the significance of the weights on the edges of a weighted graph?
What is the significance of the weights on the edges of a weighted graph?
A Hamiltonian circuit can exist in a graph with vertices of varying degree, as long as there is a sufficient number of edges.
A Hamiltonian circuit can exist in a graph with vertices of varying degree, as long as there is a sufficient number of edges.
What is 7 hours before 3 o'clock on a 12-hour clock?
What is 7 hours before 3 o'clock on a 12-hour clock?
16 days from Friday will be Saturday.
16 days from Friday will be Saturday.
What day of the week is 14 days after Wednesday?
What day of the week is 14 days after Wednesday?
Using modular arithmetic, 29 ≡ 8 mod ______.
Using modular arithmetic, 29 ≡ 8 mod ______.
Match the following operations with their results on a 12-hour clock:
Match the following operations with their results on a 12-hour clock:
Modular arithmetic can be used to represent day-of-the-week calculations.
Modular arithmetic can be used to represent day-of-the-week calculations.
Calculate 15 - 8 mod 12.
Calculate 15 - 8 mod 12.
If today is Monday, then 9 days from now will be a ______.
If today is Monday, then 9 days from now will be a ______.
What is a key characteristic of a planar graph?
What is a key characteristic of a planar graph?
The Utilities Graph can be redrawn in a way that does not have any edge crossings.
The Utilities Graph can be redrawn in a way that does not have any edge crossings.
What is the term for a graph that is derived from another graph using its edges and vertices?
What is the term for a graph that is derived from another graph using its edges and vertices?
If a graph can be redrawn with no edges crossing, it is classified as a ______ graph.
If a graph can be redrawn with no edges crossing, it is classified as a ______ graph.
Match each graph property with its correct description:
Match each graph property with its correct description:
Which of the following statements about a subgraph is true?
Which of the following statements about a subgraph is true?
An equivalent form of a non-planar graph can still be classified as non-planar if it has crossing edges.
An equivalent form of a non-planar graph can still be classified as non-planar if it has crossing edges.
List one strategy to determine if a graph is not planar.
List one strategy to determine if a graph is not planar.
A graph that has edges which connect opposite vertices but require crossing is considered ______.
A graph that has edges which connect opposite vertices but require crossing is considered ______.
Which property makes the Utilities Graph non-planar?
Which property makes the Utilities Graph non-planar?
Flashcards
Closed Path or Circuit
Closed Path or Circuit
A path that starts and ends at the same vertex.
Euler Circuit
Euler Circuit
A circuit that uses every edge exactly once and never uses the same edge twice.
Degree of a Vertex
Degree of a Vertex
The number of edges that meet at a vertex.
Eulerian Graph
Eulerian Graph
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Non-Eulerian Graph
Non-Eulerian Graph
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Open Path
Open Path
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Non-Eulerian Graph Theorem
Non-Eulerian Graph Theorem
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Trial and Error
Trial and Error
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Planar Graph
Planar Graph
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Planar Drawing
Planar Drawing
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Subgraph
Subgraph
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Hamiltonian Circuit
Hamiltonian Circuit
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Subgraph Method
Subgraph Method
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The Utilities Graph
The Utilities Graph
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Non-Planar Graph
Non-Planar Graph
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Redrawing
Redrawing
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Map Coloring Problem
Map Coloring Problem
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Chromatic Number
Chromatic Number
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Four-Color Theorem
Four-Color Theorem
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Trial and Error Coloring
Trial and Error Coloring
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3-Colorable Graph
3-Colorable Graph
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Modular Arithmetic
Modular Arithmetic
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Arithmetic modulo n
Arithmetic modulo n
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Result of Modulo n
Result of Modulo n
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Addition Modulo n
Addition Modulo n
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Subtraction Modulo n
Subtraction Modulo n
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Clock Arithmetic
Clock Arithmetic
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Days of the Week Arithmetic
Days of the Week Arithmetic
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Calculating Times
Calculating Times
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ISBN (International Standard Book Number)
ISBN (International Standard Book Number)
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ISBN Check Digit
ISBN Check Digit
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Modulo n Operation
Modulo n Operation
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Modulus (in Modular Arithmetic)
Modulus (in Modular Arithmetic)
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ISBN Check Digit Calculation
ISBN Check Digit Calculation
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Transposition Errors
Transposition Errors
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ISBN Check Digit Verification
ISBN Check Digit Verification
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Modular Arithmetic in Error Detection
Modular Arithmetic in Error Detection
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Modulo (mod)
Modulo (mod)
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Congruence (≡)
Congruence (≡)
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Day-of-the-Week Arithmetic
Day-of-the-Week Arithmetic
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Modular Notation (a ≡ b mod n)
Modular Notation (a ≡ b mod n)
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Reducing Modulo n
Reducing Modulo n
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Modular Operations
Modular Operations
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Vertex
Vertex
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Edge
Edge
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Euler's Circuit Theorem
Euler's Circuit Theorem
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Euler's Formula
Euler's Formula
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Face in a Planar Graph
Face in a Planar Graph
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Modulus (n)
Modulus (n)
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What is an ISBN?
What is an ISBN?
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What is an ISBN Check Digit?
What is an ISBN Check Digit?
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What is Modular Arithmetic?
What is Modular Arithmetic?
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What is a UPC?
What is a UPC?
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What is the Luhn Algorithm?
What is the Luhn Algorithm?
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What is Clock Arithmetic?
What is Clock Arithmetic?
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What is Day-of-the-Week Arithmetic?
What is Day-of-the-Week Arithmetic?
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What is a Transposition Error?
What is a Transposition Error?
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Euler Path
Euler Path
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What is a Complete Graph?
What is a Complete Graph?
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Greedy Algorithm
Greedy Algorithm
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Edge-Picking Algorithm
Edge-Picking Algorithm
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Weight of a Circuit
Weight of a Circuit
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Weighted Graph
Weighted Graph
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Starting Vertex
Starting Vertex
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Finding the 'Cheapest' Circuit
Finding the 'Cheapest' Circuit
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Algorithms Aren't Perfect
Algorithms Aren't Perfect
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Total Weight of a Hamiltonian Circuit
Total Weight of a Hamiltonian Circuit
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Travel Cost in Weighted Graphs
Travel Cost in Weighted Graphs
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Hamiltonian Circuit for Travel Cost
Hamiltonian Circuit for Travel Cost
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Flight Cost in Weighted Graphs
Flight Cost in Weighted Graphs
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Greedy Algorithm Circuit Weight
Greedy Algorithm Circuit Weight
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Edge Picking Algorithm Circuit Weight
Edge Picking Algorithm Circuit Weight
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Loop
Loop
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Connected Graph
Connected Graph
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Complete Graph
Complete Graph
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Equivalent Graphs
Equivalent Graphs
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Path in a Graph
Path in a Graph
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Circuit
Circuit
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Weight of a Hamiltonian Circuit
Weight of a Hamiltonian Circuit
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Traveling Salesperson Problem
Traveling Salesperson Problem
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Clock Addition Symbol (⊕)
Clock Addition Symbol (⊕)
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Clock Subtraction Symbol (⊖)
Clock Subtraction Symbol (⊖)
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What is Modulo (mod)?
What is Modulo (mod)?
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Study Notes
Introduction to Graphs
- Graph theory is a branch of mathematics that analyzes connections.
- Graphs are diagrams with vertices (points) and edges (lines or curves) connecting vertices.
- Vertices represent entities, and edges illustrate relationships between them.
- Graphs are used to model various connections, such as friendships on social media, roads between cities, and the internet.
Example 1 - Constructing a Graph
- A table shows which students study together.
- "X" indicates two students study in the same group.
- Vertices represent students.
- Edges connect vertices if corresponding students study together.
Example 1 - Solution
- A graph shows how students are connected.
- The student who participates in the most study groups is identified through the number of connections to other students.
- The student with one connection is identified as someone who forms only one study group.
Introduction to Graphs (continued)
- A graph can include vertices without edges or multiple edges connecting the same vertices; and an edge that is connected to the vertex (a loop).
- A connected graph means any vertex can reach any other vertex by following edges.
- In a complete graph, every possible edge is drawn between all vertices.
- Graphs can represent various relationships and connections in real-world scenarios.
Euler Circuits
- Euler circuits are paths that use every edge exactly once in a graph.
- Euler circuits are called Eulerian.
- The graph must have all even degrees of vertices.
- The Königsberg bridges problem is solved by representing the land regions as vertices and connecting two vertices if a bridge exists between them.
- Euler circuits are important in solving problems involving traversing edges exactly once.
Euler Paths
- Euler paths are paths that use every edge exactly once (without returning to the starting point).
- The graph must have exactly two vertices of odd degree.
- The Euler path must start at one odd degree vertex and end at the other point of odd degree vertex.
- Euler paths are used for route planning and traversing networks.
Weighted Graphs
- A weighted graph has edges with values (weights), which can represent any quantity (e.g., distance, cost).
- For Hamiltonian circuits in a weighted graph, summing the weights along the traversed edges yields the total distance traveled.
- The traveling salesman problem seeks the shortest total distance for a Hamiltonian circuit, where the goal is to travel to all locations and return to the starting point in the shortest possible way.
- Weighted graphs are used to model problems with various quantities associated with relationships.
Hamiltonian Circuits
- A Hamiltonian circuit is a path that visits every vertex exactly once and returns to the starting vertex.
- Dirac's theorem gives conditions for a graph to contain a Hamiltonian circuit: the connected graph must have at least 3 vertices, and every vertex has a degree equal to or greater than n/2, where n is the number of vertices.
- The greedy algorithm and edge-picking algorithm are strategies for finding good Hamiltonian circuits. These methods can find a solution that might not be the absolute best but one that is "pretty good" or a "good solution" based on the given constraints.
- Hamiltonian circuits are vital in problems like finding optimal routes or tours through a network of locations.
Planarity
- A planar graph can be drawn on a plane so that no edges intersect.
- The utilities graph puzzle is illustrated with three utility companies and three houses; if the pipes connect without crossing, the graph is planar; if they cross, the graph is nonplanar.
- A subgraph of a graph consists of vertices and edges from the graph.
- K5 is a complete graph with five vertices.
- The subgraph theorem states that if a graph contains a non-planar subgraph (like the Utilities Graph or K5), the graph itself is non-planar.
- Contractions in graphs provide a way to investigate nonplanar graphs and identify if such a graph is a subgraph of a larger graph.
- Planar graphs are crucial in many applications involving physical layouts, like map designs.
Euler's Formula
- Euler's formula relates the number of vertices (v), edges (e), and faces (f) of a planar graph via the relationship v + f = e + 2.
Graph Coloring
- Graph coloring is used to color each vertex of a graph such that no two connected vertices share the same color.
- The chromatic number is the minimum number of colors required to color a graph so that each edge connects vertices of different colors.
- The four-color theorem states that every planar graph is 4-colorable, so only 4 colors are required to complete such a coloring.
- Applications of graph coloring are widely varied and include scheduling problems; colorings are done so that when vertices are connected, they are not the same colors.
- Graph coloring is used in various scenarios where conflicts need to be avoided, and scheduling or design problems require careful assignment of different resources to avoid simultaneous use of the same resource.
Modular Arithmetic
- Modular arithmetic works with remainders when integers are divided.
- Modular arithmetic is used for clock arithmetic and day-of-the-week calculations; 'modulo n' refers to the repeated pattern of mathematical results based on the remainder after division.
- The concept of modulus n provides a mathematical framework for phenomena that repeat in cycles or involve cyclic patterns in mathematics.
Mathematical Systems
- Mathematical systems describe and model how a set of elements behaves with certain operations.
- Modular arithmetic is a specific system with certain rules for addition and subtractions, resulting in predictable patterns that are fundamental to many aspects of applied mathematics.
- Mathematical systems provide structure and models for various forms of data/information analysis, problem solving, and pattern recognition.
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