Prove that edge $e$ is a cut edge in graph $G$ iff $e$ is not in every circle in the graph.
Actually I am rather confusing at terms: every circle in the graph. What circle is this mean? That's makes me stuck to think about the proof of this case. I have asked some of my friends but they seem don't have any ideas for it. Could you help me?
Suppose an edge is a cut edge and it is a part of some circle. Then if we remove it than there will be still a path between vertices that where connected with that edge. So this edge can't be part of any edge.
Vice versa. If $e$ is not a part of any circle, then by removing it the vertices that where connected by this edge will not be connected, otherwise we would have a circle.