Let G be a connected graph. We say that G is minimally connected if the removal of any edge of G (without deleting any vertices) results in a disconnected graph. (a) Show that a connected, minimally connected graph has no cycles. (b) Show that a connected graph with no cycles is minimally connected. Commentary: The exercise shows that the two conditions are equivalent. A graph satisfying either condition is a tree.
The Attempt at a Solution
I approach this by proving that a tree is a minimally connected graph
A graph G is a tree if and only if every two vertices of G are connected by one path.
Let G be a tree, delete any edge.
This deleted edge connected a pair of vertices V1V2 which implies that the only path between V1 and V2 was the deleted edge. This shows that there is no path from V1 to V2. Therefore, the graph becomes disconnected hence we can't get from any vertex to any vertex.
The assignment states that G is minimally connected if the removal of any edge of G results in a disconnected graph. Based on the definition of a tree and the stated proof we can say that a tree is a minimally connected graph and therefore it will satisfy the two previous conditions.
Is that right?