Question

Prove that Reverse-Delete produces a spanning tree of G

a. Counterexample exists
b. Proof by contradiction
c. By mathematical induction
d. Direct proof

Answer 1

The Reverse-Delete algorithm produces a minimal spanning tree by sorting edges in decreasing weight, deleting unnecessary edges, and keeping only those that are essential for maintaining graph connectivity, thus ensuring no cycles are formed and all vertices stay connected.

Step-by-step explanation:

To prove that the Reverse-Delete algorithm produces a spanning tree of a graph G, we apply a direct proof. The Reverse-Delete algorithm works as follows:

1. Sort the edges in decreasing order of weight.
2. Iterate through each edge in this order, and for each edge, check if its removal disconnects the graph.
3. If removing the edge would disconnect the graph, retain the edge. Otherwise, delete it.

In the end, the Reverse-Delete algorithm will result in a spanning tree because during the process, it never deletes an edge that would disconnect the graph, meaning all vertices remain connected. Moreover, it will not leave a cycle in the graph because if an edge can be removed without disconnecting the graph, it means that there is an alternative path between the vertices, which would have formed a cycle.

The result is a minimal spanning tree because the algorithm begins with the heaviest edges, which would contribute to the highest total weight. By attempting to eliminate these edges first—and only keeping them if necessary for maintaining connectivity—the algorithm ensures that the remaining edges compose a tree of minimal total weight.