Let G be an arbitrary graph with at least 2 vertices. We construct a graph H by adding two vertices x and y to G, with every possible edge between vertices of G and x, y. Prove that H will never have any cut vertices, no matter what graph G we start with.

Answer 1

Cannot graph on these posts.

Answer 2

This article should give you a good starting point.

www.tutorialspoint.com/graph_theory/graph_theory_connectivity.htm