Is maximal independent set of a graph unique? I think between indepent sets, only one of them is maximal. So does it prove that MIS is unique?
2026-03-30 15:34:50.1774884890
uniqueness of Maximal Independent Set(MIS)
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Maximal independent sets are not unique. Consider $Q_{3}$, the hypercube on $8$ vertices. I can select two non-adjacent vertices and up to four non-adjacent vertices depending on my selection. Each such selection is a maximal independent set.
The Wikipedia page has a really nice graphic demonstrating this: http://en.wikipedia.org/wiki/File:Cube-maximal-independence.svg
Note that maximal is not the same as maximum. A maximal independent set is an independent set such that the addition of a vertex will render the set no longer independent. A maximum independent set is a maximal independent $S$ set such that for all other maximal independent sets $X$, $|S| \geq |K|$. So we can have multiple maximum independent sets, but the cardinality is unique.