1. Let G(V, ?, w) be an undirected weighted graph, that is, let V be the graph's set of nodes,
let E be the graph's set of edges, and let w: E \rightarrow \mathbb{R}^+ be the function that assigns positive
weights to edges. (For instance, V = \{1,2,3\}, E = \{\{1,2\}, \{2,3\}\}, w(\{1,2\}) = 1.5 and
w(\{2,3\}) = 1.)
Consider the following claim: "For any two nodes $v, v'$ of V, if $\pi$ is a shortest path in
G(V, E, w) from v to $v'$, then $\pi$ is a shortest path from v to $v'$ even in the graph G(V, E, w'),
where w'(e) = w(e)$^2$ for each e \in E. (That is, the sequence of nodes of a shortest path does
not change, if we square the weight of the edges.)"
Your task is to determine whether the claim is true or false. I.e., either prove the claim, or
give a counterexample.
