Homework problem O on Minimum Spanning Trees

Due Nov 11.

Safe yet heavy.

  1. (4 points) Exercise 23.1-2

    This exercise is to give a counterexample to the converse of Theorem 23.1. You may present your counterexample graph by diagram or other means. Explain how it is a counterexample to the statement clearly. Be sure the cut and the specific edge are clearly identified.

  2. (6 points) Exercise 23.1-6

    Paraphrase: For a connected, weighted, undirected graph G: Every cut has unique lightest edge ==> G has a unique MST. (but not conversely)

    Hint: the proof of theorem 23.1 is relevant.

Remark: An easy corollary of this is that if the graph has distinct weights (no two edges have the same weight), then the minimal spanning tree is unique.