Question

consider a weighted graph in which all the weights on the edges are distinct. suppose that e is the edge with the largest weight. what is the property that the graph must have in order for e to be in the minimum spanning tree?

Answer 1

For the edge with the largest weight to be in the minimum spanning tree, the graph must be structured such that this edge is part of the only path connecting two vertices, honoring the cut property and maintaining graph connectivity without forming a cycle.

Step-by-step explanation:

• To answer the question: In order for edge e with the largest weight to be included in the minimum spanning tree (MST) of a weighted graph, the graph must be structured such that e is part of the only path connecting two specific vertices in the graph.
• This concept is derived from the cut property of MSTs, which states that if you partition (cut) the vertices of the graph into two nonempty subsets, the edge with the smallest weight that connects the two subsets must be part of the MST, unless its inclusion would form a cycle.

• If e is the heaviest edge and part of the MST, its inclusion indicates that removing e would either increase the overall weight of the MST or disconnect part of the graph, meaning there is no alternative lighter edge that can connect those parts of the graph without forming a cycle.
• Therefore, this must be the unique situation where the heavier edge is necessary to maintain the connectivity of the graph.