(i) Prove that a graph is 2-edge-connected if and only if any two distinct vertices are joined by at least two paths with no edges in common. (ii) Prove that a graph with at least three vertices is 2-connected if and only if any two distinct vertices are joined by at least two paths with no other vertices in common.
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First, we prove that if a graph is 2-edge-connected, then any two distinct vertices are joined by at least two paths with no edges in common. Suppose G is a 2-edge-connected graph. Let u and v be any two distinct vertices in G. Since G is 2-edge-connected, there Show more…
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