# graph with a cut vertex contains a bridge

Give a counter example to each of the following:
(a) G is a connected graph with a cut-vertex, then G contains a bridge. (b) G is a tree if and only if it contains no cycle.

– user694818
Nov 19, 2019 at 14:17

For $$(a)$$, there are no counterexamples. this is a consequence of vertex connectivity being less than or equals to the edge connectivity.
For $$(b)$$, if we don't assume $$G$$ connected, any forest (union of trees) with more than one component will work. Note the definition of tree is a connected graph with no cycles.