Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and edge e of G, v and e lie on a common cycle of G.
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- Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if for every triple (x, y, z) of distinct vertices and any edge e not incident with y, G has an x, z-path through e that does not contain y.Let G be a connected graph of order n and size n. Prove that G contains a single cycle.