GATE | GATE-CS-2015 (Set 1) | Question 65

Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is _______________.
(A) 24
(B) 20
(C) 32
(D) 64

Answer: (A)
Explanation: Euler’s formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, then

    v − e + f = 2.  
    v -> Number of vertices
    e -> Number of edges
    f -> Number of faces

As per the question 
v = 10
And number of edges on each face is three
Therefore, 2e = 3f  [Note that every edge is
                    shared by 2 faces]

Putting above values in v − e + f = 2
10 - e + 2e/3 = 2
e = 3*10 - 6 = 24

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