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Geometric Graph Theory

Description: Geometric Graph Theory Quiz: Test Your Knowledge of Geometric Properties of Graphs
Number of Questions: 14
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Tags: graph theory geometric graph theory mathematics
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In a geometric graph, what is the maximum number of edges that can be drawn between any two vertices?

  1. 1

  2. 2

  3. 3

  4. 4

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Correct Option: A
Explanation:

In a geometric graph, each edge represents a straight line segment between two vertices. Since two points can only be connected by one straight line, the maximum number of edges between any two vertices is 1.

What is the minimum number of edges required to connect all vertices in a geometric graph?

  1. n-1

  2. n

  3. n+1

  4. 2n-1

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Correct Option: A
Explanation:

In a geometric graph, the minimum number of edges required to connect all vertices is n-1, where n is the number of vertices. This is because a tree, which is a connected graph with no cycles, can be constructed with n-1 edges.

What is the maximum number of edges that can be drawn in a geometric graph with n vertices?

  1. n(n-1)/2

  2. n(n+1)/2

  3. n^2

  4. 2n^2

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Correct Option: A
Explanation:

In a geometric graph, the maximum number of edges that can be drawn is n(n-1)/2, where n is the number of vertices. This is because each vertex can be connected to at most n-1 other vertices.

What is the diameter of a geometric graph?

  1. The length of the longest path between any two vertices

  2. The length of the shortest path between any two vertices

  3. The number of vertices in the graph

  4. The number of edges in the graph

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Correct Option: A
Explanation:

The diameter of a geometric graph is the length of the longest path between any two vertices. It is a measure of how far apart the vertices in the graph are.

What is the radius of a geometric graph?

  1. The length of the longest path between any two vertices

  2. The length of the shortest path between any two vertices

  3. The number of vertices in the graph

  4. The number of edges in the graph

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Correct Option: B
Explanation:

The radius of a geometric graph is the length of the shortest path between any two vertices. It is a measure of how close together the vertices in the graph are.

What is the girth of a geometric graph?

  1. The length of the longest path between any two vertices

  2. The length of the shortest path between any two vertices

  3. The length of the shortest cycle in the graph

  4. The number of vertices in the graph

Show answer
Correct Option: C
Explanation:

The girth of a geometric graph is the length of the shortest cycle in the graph. It is a measure of how long the cycles in the graph are.

What is the chromatic number of a geometric graph?

  1. The minimum number of colors required to color the vertices of the graph so that no two adjacent vertices have the same color

  2. The maximum number of colors required to color the vertices of the graph so that no two adjacent vertices have the same color

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The chromatic number of a geometric graph is the minimum number of colors required to color the vertices of the graph so that no two adjacent vertices have the same color. It is a measure of how difficult it is to color the graph.

What is the independence number of a geometric graph?

  1. The maximum number of vertices that can be selected from the graph such that no two selected vertices are adjacent

  2. The minimum number of vertices that can be selected from the graph such that no two selected vertices are adjacent

  3. The number of vertices in the graph

  4. The number of edges in the graph

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Correct Option: A
Explanation:

The independence number of a geometric graph is the maximum number of vertices that can be selected from the graph such that no two selected vertices are adjacent. It is a measure of how many vertices in the graph can be colored with the same color.

What is the clique number of a geometric graph?

  1. The maximum number of vertices that can be selected from the graph such that all selected vertices are adjacent

  2. The minimum number of vertices that can be selected from the graph such that all selected vertices are adjacent

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The clique number of a geometric graph is the maximum number of vertices that can be selected from the graph such that all selected vertices are adjacent. It is a measure of how many vertices in the graph can be colored with the same color.

What is the domination number of a geometric graph?

  1. The minimum number of vertices that need to be selected from the graph such that every other vertex in the graph is adjacent to at least one selected vertex

  2. The maximum number of vertices that need to be selected from the graph such that every other vertex in the graph is adjacent to at least one selected vertex

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The domination number of a geometric graph is the minimum number of vertices that need to be selected from the graph such that every other vertex in the graph is adjacent to at least one selected vertex. It is a measure of how difficult it is to control the graph.

What is the total domination number of a geometric graph?

  1. The minimum number of vertices that need to be selected from the graph such that every other vertex in the graph is adjacent to at least two selected vertices

  2. The maximum number of vertices that need to be selected from the graph such that every other vertex in the graph is adjacent to at least two selected vertices

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The total domination number of a geometric graph is the minimum number of vertices that need to be selected from the graph such that every other vertex in the graph is adjacent to at least two selected vertices. It is a measure of how difficult it is to totally control the graph.

What is the connected domination number of a geometric graph?

  1. The minimum number of vertices that need to be selected from the graph such that the subgraph induced by the selected vertices is connected and every other vertex in the graph is adjacent to at least one selected vertex

  2. The maximum number of vertices that need to be selected from the graph such that the subgraph induced by the selected vertices is connected and every other vertex in the graph is adjacent to at least one selected vertex

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The connected domination number of a geometric graph is the minimum number of vertices that need to be selected from the graph such that the subgraph induced by the selected vertices is connected and every other vertex in the graph is adjacent to at least one selected vertex. It is a measure of how difficult it is to connect the graph and control it.

What is the path cover number of a geometric graph?

  1. The minimum number of paths that need to be selected from the graph such that every vertex in the graph is on at least one selected path

  2. The maximum number of paths that need to be selected from the graph such that every vertex in the graph is on at least one selected path

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The path cover number of a geometric graph is the minimum number of paths that need to be selected from the graph such that every vertex in the graph is on at least one selected path. It is a measure of how difficult it is to cover the graph with paths.

What is the cycle cover number of a geometric graph?

  1. The minimum number of cycles that need to be selected from the graph such that every vertex in the graph is on at least one selected cycle

  2. The maximum number of cycles that need to be selected from the graph such that every vertex in the graph is on at least one selected cycle

  3. The number of vertices in the graph

  4. The number of edges in the graph

Show answer
Correct Option: A
Explanation:

The cycle cover number of a geometric graph is the minimum number of cycles that need to be selected from the graph such that every vertex in the graph is on at least one selected cycle. It is a measure of how difficult it is to cover the graph with cycles.

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