Question

Euler's formula (v – e + f = 2) holds for all connected planar graphs. What if a graph is not connected? Suppose a planar graph has two components. What is the value of v- e + f now? What if it has k components?

Answer 1

Euler's formula for a connected planar graph is v - e + f = 2. For graphs with multiple disconnected components, the formula is modified to v - e + f = 2k, where k is the number of components.

Step-by-step explanation:

Euler's formula, which states that for any connected planar graph the relationship v - e + f = 2 holds, where v is the number of vertices, e is the number of edges, and f is the number of faces, including the outer infinite face.

When the graph is not connected, say it has k components, the equation needs to be adjusted to account for each of those components. The modified Euler's formula becomes v - e + f = 2k. For example, if a planar graph has two components, the value would be v - e + f = 4.

This is because each component adds one more to the overall expected count, assuming each component is like a separate connected planar graph.