(1) Let T be a minimum spanning tree of a weighted graph G. Construct a new graph G' by adding a weight of k to every edge of G. Do the edges of T form a minimum spanning tree of G'? Prove the statement or give a counterexample.

Ans: In a graph with n vertices, every spanning tree has n-1 edges. Thus the weight of every spanning tree is increased by exactly (n-1)k. The set of minimal weight spanning trees remains the same.

A rising tide lifts all boats.

(2) Let P = {s, ... , t} describe a shortest weighted path between vertices s and t of a weighted graph G. Construct a new graph G' by adding a weight of k to every edge of G. Does P describe a shortest path from s to t in G'? Prove the statement or give a counterexample.

Ans: Not true because adding k to each edge increases the weight of longer paths more than shorter ones. For instance, in G the shortest path a to c is via b, but in G' (k = 2 added to each edge) the direct link (a,c) is shorter.

G:                     G': 
   a                       a
1 / \ 3                 3 / \ 5
 /   \                   /   \
b --- c                 b --- c
   1                       3

A rising tide lifts more in long boat trains than in short boat trains.