Prove that Eulers formula fails for disconnected graphsSolut

Prove that Euler’s formula fails for disconnected graphs.

Solution

Euler\'s Formula (n - e + f = 2) is the basic counting tool relating vertices,edges, and faces in planar graphs.

Euler\'s Formula as stated fails for disconnected graphs. If a plane graph
G has k components, then adding k-1 edges to G yields a connected plane graph
without changing the number of faces. Hence Euier\'s Formula generalizes for
plane graphs with k components as n - e + f = k + 1 (for example, consider a
graph with n vertices and no edges).