40 40 votes 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 $3$ $4$ $5$ $6$ Graph Theory gatecse-2012 graph-theory graph-planarity normal + – gatecse 16.0k views answer comment Share Follow Print See all 4 Comments 4 4 Comments reply Rakesh K commented Dec 5, 2016 reply Follow flag Does bounded faces mean cycles? 0 0 replyShare Nisha Bharti commented Jan 12, 2023 reply Follow flag What is bounded faces? 0 0 replyShare Thadymademe commented Sep 6, 2023 reply Follow flag @Rakesh K @Nisha Bharti , Bounded faces means interior faces . In any connected planar graph , number of exterior faces is 1 Total number of faces = Total number of interior faces + total number of exterior faces , Total number of faces = Total number of interior faces + 1 Therefore , Total number of interior faces = Total number of faces – 1 5 5 replyShare Het2312 commented Apr 8 reply Follow flag Question became more intresting with 7 as option :) 2 2 replyShare Please log in or register to add a comment.
Best answer 55 55 votes For any planar graph, $\text{n(no. of vertices) - e(no. of edges) + f(no. of faces) = 2}$ $f = 15 - 10 + 2= 7$ number of bounded faces $= \text{no. of faces -1}$ $= 7 -1=6$ So, the correct answer would be D admin answered Aug 6, 2014 • edited Mar 19, 2018 by sourav. admin comment Share Follow See all 6 Comments 6 6 Comments reply Prasanna commented Nov 21, 2015 reply Follow flag "number of bounded faces = no. of faces -1" is this formula ? 0 0 replyShare LeenSharma commented Dec 1, 2015 reply Follow flag number of bounded faces = no. of faces -1 (external or unbounded face) 9 9 replyShare jugnu1337 commented Jan 15, 2022 reply Follow flag formula is vertices – edges+ faces=2 menace 10-15 while you have done edges – vertices am i correct 2 2 replyShare Kabir5454 commented Jan 15, 2022 reply Follow flag @jugnu1337 you are correct but but they asked faces so by your formula , vertices-edge + faces =2 or ,faces=edge-vertices+2 0 0 replyShare zenkirtan commented Sep 27, 2025 reply Follow flag Faces = regions Bounded = interior regions All graphs graph have 1 exterior region/face 0 0 replyShare Thadymademe commented Dec 27, 2025 reply Follow flag The first line of the answer should have been " For any connected planar graph" . The general formula is : no. of vertices + no. of faces = no. of edges + no. of components + 1 . 1 1 replyShare Please log in or register to add a comment.
2 2 votes For any planar graph v-e+r = 2 10-15+r = 2 -5 + r = 2 r= 7 number of bounded faces = no. of faces -1 (external or unbounded face) number of bounded faces = 7 – 1 = 6 akshay_123 answered Sep 3, 2023 akshay_123 comment Share Follow 0 reply Please log in or register to add a comment.
0 0 votes Number of edges in minimally connected graph: n-1 So, 10-1=9 (edges used to connect all vertices) Remaining 15-9=6 edges can be used to connect any two vertices and form a bounded face. So ans - (d) 6 Is this analogy correct? w4rb0y answered Jan 11, 2018 w4rb0y comment Share Follow 0 reply Please log in or register to add a comment.
0 0 votes Apply the Euler formula n-e+f =2 bounded face = f-1 Surya013 answered Sep 25, 2025 Surya013 comment Share Follow 0 reply Please log in or register to add a comment.
