GATE IT 2008 | Question: 4

Question

What is the size of the smallest $\textsf{MIS}$ (Maximal Independent Set) of a chain of nine nodes?

• A. $5$
• B. $4$
• C. $3$
• D. $2$

Comments

no

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you cant take 4,8 as maximal becoz adding 1 or 6 will still provide a maximal independent set

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These questions should be in graph theory and not in DS.

Answer 1

Answer: C

$1-2-3-4-5-6-7-8-9$

$(2,5,8)$ is the maximal independent set for a chain of $9$ nodes. If we add any other node to the set then it will not be MIS.

Comments

She is the best in Graph theory

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They have asked for chain …,  otherwise we can also go with 2 nodes only …?                                               please exaplined ..

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Think the best, forget the rest

Answer 2

1--2--3--4--5--6--7--8--9

MIS of size 5 : 1,3,5,7,9

MIS os size 4 : 2,4,6,8

MIS of size 3 : 2,5,8

MIS is independent set in which If we add any other node (outside MIS) to the set then it will  be no more MIS.

so smallest size of MIS is 3   Option C

Comments

@AKASH G more can exist 

but here example which u have given, can't take that, as we can include here 9 too,  so size will become 4 not 3

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If you have somemore question related to this or resources please add . 

@MANSI_SOMANI got it 👍 thanks! 

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How set {1,9} will not be a MIS for the above sequence?

Answer 3

If we try to add any vertext in set {2,5,8} then that vertex will be adjacent to 2 or 5 or 8 like if we want to add 4 that will be adjacent to 5 so independent set property will be lost.We cant take 2 vertices as Maximal property will not hold, we have to make it mamximum in such a way that the no of vertices will be minimum to make it MIS.1,4,7 is also a MIS

so C) is the ans

Comments

why cant we take only 2 vertices only... i mean just 2 and 7 or 2 and 9 ...?

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@pawan Because then we can include other vertices, so it's not MIS.

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wrong in {1,4,7} we can add 9 to make it mis.

Answer 4

A set of vertices is called independent set such that no two vertices in the set are adjacent. A maximal independent set (MIS) is an independent set which is not subset of any other independent set.

The question is about smallest MIS. We can see in below diagram, the three highlighted vertices (2nd, 5th and 8th) form a maximal independent set (not subset of any other MIS) and smallest MIS.

0----0----0----0----0----0----0----0----0

Comments

why cant we take only 2 vertices only... i mean just 2 and 7 or 2 and 9 ...?

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here maximal independent set referes to smallest independent set if you add any other vertex to it it would no longer be independent set

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here we cant add (2,7) or (2,9) because in case of (2,7) we can add 9 which is not adjacent so it will not become MIS anymore and in case of (2,9( also we can add 5 or 7 so it will not be MIS.

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I got the answer after thinking about this point, MIS means if you add any(single) node, it should be no longer independent set(IS) (should break IS rule), but in case of 2 vertices, by adding a node IS rule is not breaking.