Question

A connected planar graph having 6 vertices, 7 edges contains _______ regions.

(a) 15
(b) 3
(c) 1
(d) 11
​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{A connected planar graph having 6 vertices}$

$\textsf{and 7 edges}$

$\underline{\textbf{To find:}}$

$\textsf{Number of regions}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Euler's formula:}}$

$\boxed{\mathsf{|V|-|E|+|F|=2}}$

$\mathsf{where}$

$\mathsf{|V|-Number\;of\;vertices}$

$\mathsf{|E|-Number\;of\;edges}$

$\mathsf{|F|-Number\;of\;regions}$

$\textsf{As per given data,}$

$\mathsf{|V|=6\;\;and\;\;|E|=7}$

$\mathsf{|V|-|E|+|F|=2}$

$\implies\mathsf{6-7+|F|=2}$

$\implies\mathsf{-1+|F|=2}$

$\implies\boxed{\mathsf{|F|=3}}$

$\therefore\textbf{The given graph will contain 3 regions}$

$\underline{\textbf{Answer:}}$

$\textsf{Option (b) is correct}$