Maximal independent set

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.

For example, in the graph ${\displaystyle P_{3}}$, a path with three vertices ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, and two edges ${\displaystyle ab}$ and ${\displaystyle bc}$, the sets ${\displaystyle \{b\}}$ and ${\displaystyle \{a,c\}}$ are both maximally independent. The set ${\displaystyle \{a\}}$ is independent, but is not maximal independent, because it is a subset of the larger independent set ${\displaystyle \{a,c\}}$. In this same graph, the maximal cliques are the sets ${\displaystyle \{a,b\}}$ and ${\displaystyle \{b,c\}}$.

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