26 votes 26 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 asked Aug 5, 2014 • edited Jan 12, 2023 by gatecse gatecse 10.0k views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply Rakesh K commented Dec 5, 2016 reply Follow Share Does bounded faces mean cycles? 0 votes 0 votes Nisha Bharti commented Jan 12, 2023 reply Follow Share What is bounded faces? 0 votes 0 votes abir_banerjee commented Sep 6, 2023 reply Follow Share @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 2 votes 2 votes Please log in or register to add a comment.
Best answer 38 votes 38 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 4 Comments See all 4 4 Comments reply Prasanna commented Nov 21, 2015 reply Follow Share "number of bounded faces = no. of faces -1" is this formula ? 0 votes 0 votes LeenSharma commented Dec 1, 2015 reply Follow Share number of bounded faces = no. of faces -1 (external or unbounded face) 7 votes 7 votes jugnu1337 commented Jan 15, 2022 reply Follow Share formula is vertices – edges+ faces=2 menace 10-15 while you have done edges – vertices am i correct 0 votes 0 votes Kabir5454 commented Jan 15, 2022 reply Follow Share @jugnu1337 you are correct but but they asked faces so by your formula , vertices-edge + faces =2 or ,faces=edge-vertices+2 0 votes 0 votes Please log in or register to add a comment.
1 votes 1 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 See all 0 reply Please log in or register to add a comment.
0 votes 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 See all 0 reply Please log in or register to add a comment.