1. Suppose that (u, v) is an edge of graph G such that G-(u, v) is connected. Pr

Question

1. Suppose that (u, v) is an edge of graph G such that G-(u, v) is connected. Prove that (u, v) must belong to a cycle in G. 2. Prove that if u and v are bit strings of length c that differ in n bits, then Qe contains a (u, v)-path of length n, for any n21. Hint: proving this by induction on n may make the problem easier than proving for an arbitrary n. 3. Use the formal definition of isomorphism to prove that if G H, with iso- morphism f, and P: P, P2, ...,Pk is a path in G, then Q: f (p),f(p2)... ,f(Pk) is a path in H. 4. Use the result of the previous problem to prove that if G H, with

Explanation / Answer

Hi,
Its against chegg policy to post multiple questions as one, please post others as separate questions.
1.We can prove this using proof by contradiction.
GIiven in grapg G (u,v) is an edge such that even if we remove (u,v) edge, G is connected,
Assume there is no cycle with (u,v) edge in it, that means all paths which have (u,v) should have a common vertex v, now if we remove 'v' then the graph becomes disconnected which is a contradiction, hence our assumption is wrong and (u,v) is part of a cycle
Thumbs up if this was helpful, otherwise let me know in comments

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