Index

Welcome to the study!

In this study, you will be exposed to the fundamentals of graph theory. No prior knowledge is required to perform this task. You will start by answering a short quiz about this topic, then watch a video about it, and finally answer a second quiz about this same topic.

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Note: You can watch the video only ONCE, so please use play, pause, fast forward and rewind to navigate the video to learn as much as you can about motion and kinetics. Scroll over the horizontal bar at the bottom of the video window to fast forward and rewind. You have at most 20 minutes to watch this video.

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1. In graph theory, what is the "degree" of a node/vertex?

The number of edges in the graph The number of edges connected to the vertex The number of vertices in the graph The number of paths that pass through the vertex

2. What is a defining feature of a spanning tree?

It must be a circuit that includes every vertex.

It connects all vertices in a graph using the minimum number of edges necessary, ensuring there are no circuits.

It is a path that has the highest possible total weight. It must contain at least one vertex with a degree of 3.

3. Kruskal's algorithm and the Sorted Edges algorithm both begin by sorting edges from cheapest to most expensive. How do their rules for rejecting an edge differ?

Kruskal's rejects an edge only if it forms a circuit; Sorted Edges rejects an edge if it forms a circuit OR creates a vertex of degree 3.

Sorted Edges rejects an edge only if it forms a circuit; Kruskal's rejects an edge if it forms a circuit OR creates a vertex of degree 3.

Kruskal's is used for finding paths, while Sorted Edges is used for finding circuits, so their rules are unrelated.

There is no difference in their rejection rules; they are the same algorithm.

4. For a connected graph to possess an Euler circuit (a circuit that travels every edge exactly once), what condition must be true for all of its vertices?

Every vertex must be connected to every other vertex. Every vertex must have a degree of 2. Every vertex must have an odd degree. Every vertex must have an even degree.

5. The process of "Eulerization" is performed on a graph for what purpose?

To find the shortest Hamiltonian circuit. To remove edges until the graph becomes a spanning tree. To add new vertices to make the graph complete. To strategically duplicate existing edges so that the graph gains an Euler circuit.

6. What does the "degree" of a vertex represent?

The physical distance or cost associated with the vertex. The total count of all vertices within the entire graph. The number of edges that meet at that specific vertex. The order in which the vertex is visited in a path.

7. What is the primary goal of finding a Hamiltonian circuit in a graph?

To find a path that visits every edge exactly once without repeating. To find a circuit that visits every vertex exactly once with no repeats. To identify the shortest possible path between two specific vertices. To create a sub-graph that connects all vertices without any circuits.

8. The historical "Seven Bridges of Königsberg" problem, which was famously solved by Leonhard Euler, is fundamentally a question about which of the following graph theory concepts?

Finding the shortest path through a city. Determining if a graph contains an Euler circuit. Calculating the minimum cost to connect a set of locations. Identifying if a graph is a complete graph.

9. What fundamental difference in approach distinguishes the Nearest Neighbor algorithm from the Sorted Edges algorithm?

Nearest Neighbor is a heuristic, while Sorted Edges is an optimal algorithm.

Nearest Neighbor builds one continuous path from a starting point, while Sorted Edges assembles the circuit from the cheapest individual edges across the entire graph.

Nearest Neighbor can only be used on weighted graphs, while Sorted Edges works on unweighted graphs.

Nearest Neighbor creates a spanning tree, while Sorted Edges creates a Hamiltonian circuit.

10. When using the Sorted Edges algorithm to build a Hamiltonian circuit, why is creating a "mini-circuit" (a circuit that doesn't include all vertices) not allowed?

It violates the rule that every vertex in the final circuit must have a degree of 2. It makes the total path weight too high.

Once a smaller circuit is formed, it becomes impossible to include the remaining outside vertices without breaking the circuit or visiting a vertex twice.

Mini-circuits are only allowed in Euler circuits.

11. Kruskal's algorithm for finding a minimum cost spanning tree is notable for being both "optimal" and "efficient." What does this mean?

The algorithm is very fast, but the solution it provides is only an approximation of the best answer.

The algorithm finds the absolute best solution every time, and it does so in a computationally fast manner.

The terms "optimal" and "efficient" are interchangeable and simply mean the algorithm works well.

The algorithm is optimal for small graphs but becomes inefficient for larger ones.

12. Under what specific condition would a connected graph have an Euler path but NOT an Euler circuit?

The graph must have more than two vertices with an odd degree. The graph must have zero vertices with an odd degree. All vertices in the graph must have a degree of 3. The graph must have exactly two vertices with an odd degree.

13. Why is the "Brute Force" method for solving the Traveling Salesman Problem described as "optimal" but not "efficient"?

It is the fastest method, but its answers are only estimations.

It guarantees the absolute best solution but becomes unfeasibly slow as the number of vertices increases.

It only works for graphs that are not complete, which makes it inefficient. It is an efficient algorithm, but the optimal answer it finds is often incorrect.

14. The Nearest Neighbor algorithm is often called a "greedy" algorithm. What does this "greedy" nature entail?

It chooses the next vertex to visit completely at random. It plans the entire route from start to finish before moving.

It makes the best possible choice at the current moment (the cheapest edge) without considering long-term consequences.

It tries to visit as many vertices as possible in the shortest amount of time.

15. What is the primary improvement offered by the Repeated Nearest Neighbor algorithm over the standard Nearest Neighbor algorithm?

It guarantees the optimal solution by using a more complex formula. It sorts all the edges in the graph first, which makes the process more organized.

It mitigates the risk of a poor starting choice by running the algorithm from every vertex and selecting the best outcome.

It works on disconnected graphs, whereas the standard algorithm does not.

Welcome to the study!

After reading, please answer the following questions:

Pre test

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