Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to

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Solution

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Official answer: D) 6

Graph Theory

By Euler's formula for a connected planar graph:

V - E + F = 2

Given:

Number of vertices $V = 10$
Number of edges $E = 15$

Substitute the values into Euler's formula:

10 - 15 + F = 2 \implies F - 5 = 2 \implies F = 7

The total number of faces is $F = 7$ , which includes $1$ unbounded (outer) face and the remaining bounded (interior) faces:

\text{Number of bounded faces} = F - 1 = 7 - 1 = 6

Therefore, option D is correct.

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This Graph Theory previous year question is Q17 of the GATE CSE 2012 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. This is the only 1-mark MCQ among them, and it is the only one in GATE CSE 2012.

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