GATE CS 2021 Set 1 Q.26 (CS Q.16): Graph Theory

In an undirected connected planar graph $G$ , there are eight vertices and five faces. The number of edges in $G$ is __________

Your answer

Official answer and solution

Solution

Apply Euler's formula for connected planar graphs relating vertices, edges, and faces.

For any connected planar graph, Euler's formula states:

V - E + F = 2

where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces (including the exterior unbounded face).

We are given $V = 8$ and $F = 5$ . Substituting these values into Euler's formula:

8 - E + 5 = 2

13 - E = 2 \implies E = 11

Therefore, the number of edges in $G$ is 11.

Final answer

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