Step 1

Given

Let \(\displaystyle{G}={\left({V},{E}\right)}\) be a connected simple graph, and call the edge \(\displaystyle{\left({u},{v}\right)}\in{E}\) essential if the graph obtained by removing (u,v) from G is disconnected.

Step 2

Solution

To show that G is a tree if and only if all its edges are connected.

Since, all vertices in a tree are path connected with a unique path in between them.

So removing an edge from the graph will make it disconnected or just as two distinct subgraphs.

Thus, easily conclude that G is a tree if and only if all its edges are essential.

Given

Let \(\displaystyle{G}={\left({V},{E}\right)}\) be a connected simple graph, and call the edge \(\displaystyle{\left({u},{v}\right)}\in{E}\) essential if the graph obtained by removing (u,v) from G is disconnected.

Step 2

Solution

To show that G is a tree if and only if all its edges are connected.

Since, all vertices in a tree are path connected with a unique path in between them.

So removing an edge from the graph will make it disconnected or just as two distinct subgraphs.

Thus, easily conclude that G is a tree if and only if all its edges are essential.