Question

In the network shown below, after converting it to a topological graph, which of the following statement(s) is/are TRUE? (Assume there are no pseudo-nodes) 

• A. The correct number of nodes (or vertices) are 7
• B. The correct number of edges (or links) are 9
• C. The total number of regions are 4
• D. The correct number of edges (or links) are 8

Hint:

When converting a network drawing to a topological graph: (1) mark nodes at all endpoints and true intersections; (2) edges are segments between adjacent nodes; (3) verify counts using Euler’s formula \(F=E-V+2\) (for a connected planar graph) to avoid miscounts.

Answer 1

The Correct Option is A

Count nodes (vertices). Convert to a topological graph by placing a node at every line endpoint and at every intersection (no pseudo-nodes). The figure yields \(\mathbf{V=7}\) distinct nodes.
Count edges (links). An edge is the maximal line segment between two consecutive nodes along a path. Tracing each segment between nodes gives \(\mathbf{E=9}\) edges.
Check with Euler’s formula (planar, single component). For a connected planar graph, \[ F = E - V + 2, \] where \(F\) is the number of faces (regions), including the exterior. With \(V=7\) and \(E=9\), \[ F = 9 - 7 + 2 = \mathbf{4}. \] Thus the total number of regions is \(4\). Therefore statements (A), (B), and (C) are true; (D) is false.
\[ \boxed{V=7, E=9, F=4} \]