TITLE: Relation between articulation points and bridge edges
QUESTION [5 upvotes]: What is the relation ship between articulation points and bridges of a graph.
Specifically, if there are no articulation points in a graph is it necessary that there will be no bridge edges.

REPLY [3 votes]: Suppose a graph has a bridge edge. Then consider one of its end points. By removing that vertex, we are also removing that edge and hence disconnecting the graph. So a graph has a bridge edge implies it has an articulation point.
Note that this proof has a minor flaw. It does not consider the case when one of the connected components obtained after removing the edge has a single vertex. So the above statement fails when the graph is just two vertices and an edge between them (or if the graph is disconnected, this is one connected component of it). This is in fact the only case where the above statement is false.
And we can see by simple examples that the converse, graph has articulation point implies it has bridge edge, is not true.