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

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Solution

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Official answer: 11

Graph Theory

According to Euler's formula for a connected planar graph:

V - E + F = 2

where:

$V$ is the number of vertices ( $V = 8$ )
$E$ is the number of edges
$F$ is the number of faces ( $F = 5$ )

Substituting the given values:

8 - E + 5 = 2

13 - E = 2 \implies E = 11

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

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This Graph Theory previous year question is Q16 of the GATE CSE 2021 Set 1 paper. Find more GATE CSE PYQs on Graph Theory in Engineering Mathematics, or browse all GATE CSE PYQs.

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Graph Theory in GATE CSE

Graph Theory has 21 GATE CSE PYQs across 14 papers from 2012 to 2026. 3 of them are 1-mark NAT questions like this one, and it is the only one in GATE CSE 2021 Set 1.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.