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## Homework Statement

Prove that a graph is a tree if and only if it has no cycles and the insertion of any new edge always creates exactly one cycle.

## The Attempt at a Solution

Assume that a graph G is connected and contains no vertices with a degree of zero.

So would I get my proof by proving that the graph is a tree and it is connected and has n-1 edges which proves it has no cycles and then prove that adding a edge would create a cycle making the first statement false. Does this work?