Official PYQ · 2005Q10Graphs: Connectivity, Matching, ColouringMCQ1 mark

GATE CS 2005 Q10 — Graphs: Connectivity, Matching, Colouring

Let $G$ be a simple connected planar graph with $13$ vertices and $19$ edges. Then, the number of faces in the planar embedding of the graph is:

View worked solution

Answer: B

According to Euler's Formula for any connected planar graph:

$V - E + F = 2$

Where:

$V$ is the number of vertices
$E$ is the number of edges
$F$ is the number of faces (including the outer unbounded face)

Given in the problem:

Number of vertices, $V = 13$
Number of edges, $E = 19$

Substituting these values into Euler's formula:

$13 - 19 + F = 2$

$-6 + F = 2$

$F = 8$

Therefore, the number of faces in the planar embedding of the graph is 8.

Free practice · Your answers on this page are not saved.

Published 10 Oct 2026

Report a correction

GATE CS · 2026Graphs: Connectivity, Matching, Colouring
Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime}$ or $v \in V^{\prime}$ . Let the size of the smallest vertex cover in $G$ be $k$ . Let $S$ be any vertex cover of size $k$ . For a vertex $v \in V$ , which of the following constraints …
MCQ · 2 marks
GATE CS · 2026Graphs: Connectivity, Matching, Colouring
An undirected, unweighted, simple graph $G(V, E)$ is said to be $2$ -colorable if there exists a function $c: V \rightarrow\{0,1\}$ such that for every $(u, v) \in E, c(u) \neq c(v)$ . Which of the following statements about $2$ -colorable graphs is/are true?
MSQ · 2 marks
GATE CS · 2026Graphs: Connectivity, Matching, Colouring
Let $G$ be an undirected graph, which is a path on $8$ vertices. The number of matchings in $G$ is $\_\_\_\_$ . (answer in integer)
NAT · 2 marks
GATE CS · 2026Graphs: Connectivity, Matching, Colouring
Consider a complete graph $K_{n}$ with $n$ vertices ( $n>4$ ). Note that multiple spanning trees can be constructed over $K_{n}$ . Each of these spanning trees is represented as a set of edges. The Jaccard coefficient between any two sets is defined as the ratio of the size of the intersection of the two sets to the size of the union of the two …
MCQ · 2 marks
GATE CS · 2024Graphs: Connectivity, Matching, Colouring
The number of spanning trees in a complete graph of $4$ vertices labelled $\text{A, B, C,}$ and $\text{D}$ is _________.
NAT · 1 mark
GATE CS · 2024Graphs: Connectivity, Matching, Colouring
The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic number $k$ . Which of the following statements is/are always TRUE?
MSQ · 2 marks

Browse this topic

More Graphs: Connectivity, Matching, Colouring questions

GATE CS 2005 Q10 — Graphs: Connectivity, Matching, Colouring

Related questions