Question

The vertex connectivity of any tree is

• A. one
• B. two
• C. three
• D. None of these

Hint:

Trees are minimally connected graphs with vertex connectivity 1.

Answer 1

The Correct Option is A

To determine the vertex connectivity of any tree, we first need to understand the concept of vertex connectivity. Vertex connectivity refers to the minimum number of vertices that need to be removed to disconnect the remaining vertices of the graph or make it trivial (reduce it to a single vertex). 

Here is the step-by-step explanation:

1. A tree is a special type of graph that is both connected and acyclic. It has a simple structure where the number of edges is always one less than the number of vertices, i.e., \(n - 1\) edges for \(n\) vertices.
2. In a tree, removing any single vertex results in a graph that is disconnected because each vertex, except for the leaves, contributes to the connectivity of the whole tree.
3. Thus, by definition, the vertex connectivity of a tree is always \(1\) because removing just one vertex will make the tree disconnected.

Therefore, the correct answer is one.

Each of the other options can be ruled out:

• Two: This would mean at least two vertices need to be removed to disconnect a graph, which is not true for trees.
• Three: This implies three vertices are required for disconnection, which is incorrect for trees.
• None of these: Incorrect because the correct connectivity value is explicitly known and stated above.