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I've been reading up on matroids recently, which are used in the theory of greedy algorithms. A matroid is a pair $(X, I)$ where $X$ is a set and $I \subseteq \wp(X)$ is a family of sets over $X$ called the independent sets in $X$.

It occurred to me that I'd seen the term "independent set" also used in a graph-theoretic context to refer to a set of nodes in a graph where no two nodes in the set are adjacent.

I'm not immediately seeing a connection between these two kinds of independent sets. Notably, in a matroid, all maximal independent sets are required to have the same cardinality, while in a graph theory context, it's possible for there to be many different maximal independent sets of differing cardinalities.

Is there a connection between these two concepts of "independent sets," or is the terminology just an accident of history?

templatetypedef
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    I believe it's a coincidence of notions, but I wonder if any relation exists. – yo' Mar 24 '16 at 22:17
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    The independent sets of matroids generalize sets of linearly independent vectors in a vector space. – mhum Apr 02 '16 at 01:11
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    @templatetypedef: You can define an independence system (which is weaker than a matroid) on the independent vertices of a graph $G = (V(G), E(G)$: choose $V(G)$ as ground set and the set of independent subsets of vertices of $G$ plus the empty set as the independent sets of the independence system. You are right, of course, that in a graph theory context, it's possible for there to be many different maximal independent sets of differing cardinalities.To remedy this check (e.g.) vertex-transitive graphs or other graph-types with a very high form of symmetry. – Moritz May 14 '16 at 23:49

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Expanding on @Moritz's comment, it seems like the answer comes from a related mathematical structure called an independence system, which is a pair $(X, I)$ where $X$ is a ground set, $I \subseteq \wp(A)$, and $I$ obeys the following properties:

  • $I \ne \emptyset$, and
  • $\forall S \in I. \wp(S) \subseteq I$.

The sets in $I$ are called independent sets. The set of all independent sets in a graph $G$ form an independence system, since there's always at least one independent set (namely, the empty set) and any subset of an independent set is also an independent set.

A matroid is an independence structure that also satisfies the exchange property, which is something that independent sets in a graph-theoretic sense do not obey. So in that sense, the connection between independent sets in graph theory and independent sets in matroids comes from the independence structure of a matroid, not the exchange property.

templatetypedef
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  • It is worth pointing out that arguably the clearest example of graphs for which the set of independent sets do not constitute a matroid are complete $r$-partite graphs There, each partition class is inclusion-maximally independent, and these classes can be made as unequally-sized as one wishes. Moreover, the exchange-axiom then obviously does not hold, proving that in this graph the independence complex is not a matroid. – Peter Heinig Mar 29 '18 at 04:59