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Chapter 14: Problem 44

If a graph is given, how do you determine whether or not it is a tree?

Short Answer

Expert verified
A graph is a tree if it has no cycles, is connected, and any two vertices in the graph are connected by only one simple path.

Step by step solution

01

Checking for Cycles

In the first step, check if there are any cycles within the graph. The presence of a cycle should mean the graph is not a tree. This can be done using depth-first search or breadth-first search algorithms. When carrying out the search, if you encounter a visited node again, then there's a cycle in the graph.
02

Checking for Connection

Next, verify if the graph is a connected graph; that is, every pair of vertices is connected by at least one path. The graph should not be a disconnected graph as trees are always connected.
03

Confirming Unique Paths

Finally, one must confirm that any two vertices in the graph are connected by only one path. If you find multiple paths connecting any pair of vertices, then the graph isn't a tree.

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Most popular questions from this chapter

Use a tree to model the employee relationships among the chief administrators of a large community college system: Three campus vice presidents report directly to the college president. On two campuses, the academic dean, the dean for administration, and the dean of student services report directly to the vice president. On the third campus, only the academic dean and the dean for administration report directly to the vice president.

What is a weighted graph and what are the weights?

Describe how to determine whether or not a point where two of a graph's edges cross is a vertex.

In Exercises 11-16, a graph with no loops or more than one edge between any two vertices is described. Which one of the following applies to the description? i. The described graph is a tree. ii. The described graph is not a tree. iii. The described graph may or may not be a tree. The graph has five vertices, and there is exactly one path from any vertex to any other vertex.

Although the Nearest Neighbor Method does not always give the Hamilton circuit for which the sum of the weights is a minimum, Kruskal's Algorithm always gives the spanning tree with the smallest possible total weight.
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