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GATE20074
+11
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1.6k
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Let $G$ be the nonplanar graph with the minimum possible number of edges. Then $G$ has
9 edges and 5 vertices
9 edges and 6 vertices
10 edges and 5 vertices
10 edges and 6 vertices
gate2007
graphtheory
normal
outofsyllabusnow
asked
Sep 22, 2014
in
Graph Theory
by
Kathleen
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69k
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retagged
Dec 3, 2015
by
Akash Kanase

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2
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+14
votes
Best answer
ans is B:
non planar graph with smallest number of edges is K3,3; it has 9 edges and 6 vertices
K5 is also non planar but it has 10 edges and 5 vertices.
answered
Jan 2, 2015
by
jayendra
Boss
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8.5k
points)
selected
Dec 3, 2015
by
Akash Kanase
comment
is there any other way to find non planar graph ?
answer is b or c?
@ Regina Phalange : the question is asking minimum edges thus it is B 6 vertices with 9 edges.
There is nothing specific said about k3,3 in question? Bipartite sets have a very specific condition, we cant make any assumptions here
We have to use eulars formula here, 'c' is the answer
you can draw a graph with 6 vertices and 11 edges and still it is planner....
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–2
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We can check it with Euler's formula V  e + r = 2 and 3r $\leq$ 2e.
Take each option and check whether 3r $\leq$ 2e after finding value of r using
v  e + r = 2.
if its satisfies the condition than its is planar otherwise not.
For all option its satisfies except option c
answered
Mar 17, 2017
by
Beyonder
Junior
(
733
points)
comment
No you are wrong if it not satisfies then it is non planar but if it satisfies we cant say anything about it it can be planar or nonplanar as it is in option b although option c is nonplanar but it dont have minimum possible number of edges as mentioned in the question. Moreover option b represents K3,3 which is kurotowski's minimum edge nonplanar graph so option b is correct.
yes, @neelesh you are correct
every planer graph satisfies euler's formula, if it doesnt satisifes then it is non planer for sure.
but we can not say about nonplaner graph if they satisfies this equation or not.
exactly!
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