Question

4. A connected planar graph having 12 edges, 4 regions,

contains vertices :

(a) 15

(b) 12

(c) 16

(d) 10​

Answer 1

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

$\textsf{A connected planar graph having 12 edges, 4 regions}$

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

$\textsf{Number of verrtices}$

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

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

$\textsf{Let G(V,E) be a connected planar graph.}$

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

$\mathsf{|V|\;-\;No.\;of\;vertices}$

$\mathsf{|E|\;-\;No.\;of\;edges}$

$\mathsf{|F|\;-\;No.\;of\;regions}$

$\mathsf{Here,}$

$\mathsf{|E|=12\;\;\&\;\;|F|=4}$

$\textsf{By Euler's formula,}$

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

$\implies\mathsf{|V|-12+4=2}$

$\imples\mathsf{|V|-8=2}$

$\imples\mathsf{|V|=2+8}$

$\imples\boxed{\mathsf{|V|=10}}$

$\therefore\textbf{No. of vertices is 10}$

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

$\textbf{Option (d) is correct}$