Given a weighted graph G = (V, E, w) and a source vertex s ∈ V, prove the the following properties regardless of the SSSP algorithm. 1. If π[s] ever changes after the initialization, then G contains a...

Given a weighted graph G = (V, E, w) and a source vertex s ∈ V, prove the the following properties regardless of the SSSP algorithm.

1. If π[s] ever changes after the initialization, then G contains a negative cycle through s.

2. Show that π[s] may never change after the initialization, even when G contains a negative cycle through s.

we need to prove so do not provide codes.