A spanning tree on a graph with V vertices contains how many edges?

A spanning tree on a graph with V vertices contains how many edges?

• A. V
• B. V-1
• C. E-1
• D. E

Answer: (B)

A spanning tree is a connected acyclic subgraph that includes all the vertices of the graph. For such a tree with V vertices, the number of edges is exactly V minus 1. This can be seen by starting with one vertex and then adding each new vertex with a single connecting edge; after adding V−1 vertices, you’ve added V−1 edges. So the spanning tree uses V−1 edges no matter how many edges the original graph had, as long as the graph is connected. The other counts don’t generalize: V would imply a cycle for more than one vertex, E refers to the original graph’s edges, and E−1 isn’t guaranteed to form a spanning tree.

A spanning tree is a connected acyclic subgraph that includes all the vertices of the graph. For such a tree with V vertices, the number of edges is exactly V minus 1. This can be seen by starting with one vertex and then adding each new vertex with a single connecting edge; after adding V−1 vertices, you’ve added V−1 edges. So the spanning tree uses V−1 edges no matter how many edges the original graph had, as long as the graph is connected. The other counts don’t generalize: V would imply a cycle for more than one vertex, E refers to the original graph’s edges, and E−1 isn’t guaranteed to form a spanning tree.