3. Given a plane-drawing (i.e. no crossing edges) of a connected planar graph G,

Question

3. Given a plane-drawing (i.e. no crossing edges) of a connected planar graph G, a face is a region that is enclosed by edges. For example, the following plane-drawing of K4 has 3 faces (labelled 1,2,3): (a) How many edges must a connected graph with n vertices and 1 face have? (b) By examining several planar graphs, come up with an equation that relates the number of vertices (n), the number of edges (m) and the number of faces (f) of a plane-drawing of a planar graph (c) Prove, by induction on f or otherwise, that your formula is correct Hint: What happens if you delete an edge of a plane-drawing that doesn't disconnect the graph?

Explanation / Answer

Maximum edges can be n(n-1)/2 having n vertices .

For eg here n=4

Then max no. Of edges = 6

It has 3 faces

But according to question graph with 1 face will have

edges =[N(n-1)/2] - 2

Here n=4

Edges = 4

Whose face =1

2)

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