This problem is asking to prove that for a connected graph that does not separate the plane into different regions, the number of edges is one less than the number of vertices.

Let's denote:

• $V$ as the number of vertices.
• $E$ as the number of edges.

We are tasked with proving that $E = V - 1$.

Explanation:

This problem describes a tree, which is a specific type of graph in graph theory. A tree is a connected graph with no cycles. Here's a step-by-step outline of the proof:

1. Tree Definition:

   • A tree is a connected graph with no cycles.
   • Trees do not separate the plane into different regions (as described in the problem), which means they are planar graphs without any closed loops.

2. Base Case:

   • If the graph has one vertex ($V = 1$), then it cannot have any edges ($E = 0$), so the formula $E = V - 1$ holds because $0 = 1 - 1$.

3. Adding Vertices:

   • Now consider a tree with $V$ vertices and $E = V - 1$ edges. If you add another vertex to the tree, to keep the graph connected, you must add exactly one edge. This results in $V + 1$ vertices and $E + 1$ edges. After adding this vertex, the number of edges is still one less than the number of vertices: $(E + 1) = (V + 1) - 1$.

4. Conclusion:

   • By induction, for any connected graph that is a tree, the number of edges is always one less than the number of vertices. Thus, $E = V - 1$ for any connected graph without cycles.

In summary, the graph described in the problem is a tree, and for a tree, the number of edges is always one less than the number of vertices.

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions for further exploration:

1. What are the properties of trees in graph theory?
2. How do you prove the uniqueness of paths between any two vertices in a tree?
3. Can a graph with the same number of vertices and edges be connected but not a tree?
4. How would the edge-to-vertex relationship change if the graph had cycles?
5. How does this proof relate to Euler's formula for planar graphs?

Tip: When working with trees, remember that they are minimal connected graphs with no cycles, which makes them useful in optimization problems like minimum spanning trees.