Which of the following statements about spanning trees is false?AA spanning tree of a graph G is a subgraph that includes all vertices of GBA spanning tree can be a disconnected graphCA spanning tree has n−1 edges where n is the number of vertices in the original graphDEvery connected graph has at least one spanning tree.
Question
Which of the following statements about spanning trees is false?AA spanning tree of a graph G is a subgraph that includes all vertices of GBA spanning tree can be a disconnected graphCA spanning tree has n−1 edges where n is the number of vertices in the original graphDEvery connected graph has at least one spanning tree.
Solution
The false statement is B: "A spanning tree can be a disconnected graph."
Explanation:
A. True - A spanning tree of a graph G is indeed a subgraph that includes all vertices of G. This is the definition of a spanning tree.
B. False - A spanning tree cannot be a disconnected graph. By definition, a spanning tree is a tree, and a tree is a connected graph with no cycles. Therefore, a spanning tree must be a connected graph.
C. True - A spanning tree has n−1 edges where n is the number of vertices in the original graph. This is a property of trees in general, not just spanning trees.
D. True - Every connected graph has at least one spanning tree. This is a fundamental theorem in graph theory.
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