58 votes 58 votes What is the size of the smallest $\textsf{MIS}$ (Maximal Independent Set) of a chain of nine nodes? $5$ $4$ $3$ $2$ Graph Theory gateit-2008 normal graph-connectivity + – Ishrat Jahan asked Oct 27, 2014 • edited Dec 29, 2017 by kenzou Ishrat Jahan 59.1k views answer comment Share Follow See all 9 Comments See all 9 9 Comments reply Chhotu commented Nov 1, 2017 reply Follow Share Just additional information -> MIS is NPC problem and it could be reduced to vertex cover problem. 9 votes 9 votes mehul vaidya commented Jul 28, 2018 reply Follow Share according to definition of wiki. In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property. https://en.wikipedia.org/wiki/Maximal_independent_set lets take answer and see why it is true , ans is (2,5,8) point 1 : 2 ,5,8 are independent of each other , no edge between them point 2: we can not add any other vertex , If we try to add 3 , it is adjutant to 2 , If we add 4 it is adjutant to 5 point 3 : It is smallest MIS (Maximal Independent Set) we need to understand difference between maximal and maximum and also between minimal and minimum there can be many maximal set (meaning we can not add any more vertex to it , adding more vertex will make condition unsatisfiable ) . maximum set is largest among maximum set. point 4 : here 1,3,5,7,9 is maximal sets of size 5 2,4,6,8 is maximal set of size 4 also note that 1,3,5,7 is not maximal set , as we can still add 9 to it. similarly 3,6,9 is not maximal set as we can still add 1 to it. point 5: they have asked us smallest among all maximal set hence (2,5,8) is correct answer If someone has doubt try to find answer , valid according to above point , you will realize (2,5,8) is correct answer Now try to validate all above point 24 votes 24 votes Raja Rawal commented Nov 27, 2018 reply Follow Share Which option is correct.? 0 votes 0 votes soumayan bandhu commented Dec 22, 2018 reply Follow Share @mehul vaidya make this wonderful explaination answer and just one typo,maximum set is largest maximal set . 1 votes 1 votes Registered user 48 commented Jan 2, 2019 reply Follow Share (1,5,9) can also be the MIS here? 0 votes 0 votes Kuljeet Shan commented Jan 16, 2019 reply Follow Share @mehul vaidya Smallest MIS is of size 2 that is 4,8 isn't it ? 1 votes 1 votes Nandkishor3939 commented Sep 28, 2019 reply Follow Share no 0 votes 0 votes val_pro20 commented Jan 16, 2021 reply Follow Share you cant take 4,8 as maximal becoz adding 1 or 6 will still provide a maximal independent set 0 votes 0 votes neel19 commented Feb 2, 2021 reply Follow Share These questions should be in graph theory and not in DS. 2 votes 2 votes Please log in or register to add a comment.
Best answer 52 votes 52 votes 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. Rajarshi Sarkar answered Apr 16, 2015 • edited Dec 29, 2017 by kenzou Rajarshi Sarkar comment Share Follow See all 16 Comments See all 16 16 Comments reply Show 13 previous comments shashankrustagi commented Oct 31, 2020 reply Follow Share She is the best in Graph theory 0 votes 0 votes Vinay singh commented Dec 19, 2020 reply Follow Share They have asked for chain …, otherwise we can also go with 2 nodes only …? please exaplined .. 0 votes 0 votes shashankrustagi commented Dec 20, 2020 reply Follow Share Think the best, forget the rest 0 votes 0 votes Please log in or register to add a comment.
80 votes 80 votes 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 skrahul answered Dec 30, 2015 skrahul comment Share Follow See all 14 Comments See all 14 14 Comments reply Show 11 previous comments MANSI_SOMANI commented Jan 27, 2023 reply Follow Share @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 0 votes 0 votes P C commented Jan 27, 2023 reply Follow Share If you have somemore question related to this or resources please add . @MANSI_SOMANI got it 👍 thanks! 1 votes 1 votes Chaitanya Sai commented Sep 12, 2023 reply Follow Share How set {1,9} will not be a MIS for the above sequence? 0 votes 0 votes Please log in or register to add a comment.
10 votes 10 votes 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 Pranabesh Ghosh 1 answered Nov 10, 2016 • edited Apr 19, 2017 by Pranabesh Ghosh 1 Pranabesh Ghosh 1 comment Share Follow See all 3 Comments See all 3 3 Comments reply Pawan Kumar 2 commented Sep 19, 2017 reply Follow Share why cant we take only 2 vertices only... i mean just 2 and 7 or 2 and 9 ...? 0 votes 0 votes smsubham commented Dec 23, 2017 reply Follow Share @pawan Because then we can include other vertices, so it's not MIS. 1 votes 1 votes navalkishoreb commented Nov 21, 2019 reply Follow Share wrong in {1,4,7} we can add 9 to make it mis. 1 votes 1 votes Please log in or register to add a comment.
7 votes 7 votes 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 Madhab answered Jan 20, 2017 Madhab comment Share Follow See all 4 Comments See all 4 4 Comments reply Pawan Kumar 2 commented Sep 19, 2017 reply Follow Share why cant we take only 2 vertices only... i mean just 2 and 7 or 2 and 9 ...? 1 votes 1 votes Kaluti commented Oct 15, 2017 reply Follow Share here maximal independent set referes to smallest independent set if you add any other vertex to it it would no longer be independent set 2 votes 2 votes Madhab commented Nov 4, 2017 reply Follow Share 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. 1 votes 1 votes prithvish111 commented Mar 5, 2021 reply Follow Share 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. 0 votes 0 votes Please log in or register to add a comment.