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Prove that e is not bridge if and only if e is in a cycle. (graph...

Prove that e is not bridge if and only if e is in a cycle.
(graph theory)
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Answer #1

We proof this by me method of contradiction.

Suppose e is a bridge of G and on a cycle of G. By removing e, we know there is a path between two vertices of e. Which this is contradicted by definition of bridge.

Now, suppose e lies on no cycle in G. Remove e from G, and we suppose there is a path between vertices of e after removing e, as e is not a bridge. Now, add e again. As there is a path between two vertices of e which is not through e, there is a cycle in G which is passed through e. Which is contradicted by e is not a bridge. Therefore, e is a bridge of G.

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