What is a matroid?

1. What is a matroid? A matroid M is a finite set E, with a set I of subsets of E satisfying: 1.The empty set is in I 2.If X is in I, then every subset of X is also in I 3.If U, V are in I, with |U|<|V| then there is an x in V\U (V without U) such that U union {x} is in I. Write it as M = (E, I), say. E is called the ground set of M I are the independent subsets of E A matroid is an abstraction of Kruskal's algorithm.

2. Examples of matroids Example 1 - Vectors Take E as a set of vectors, and I as the set of linearly independent subsets of vectors of E. So M = (E, I) is a matroid, known as a vectorial matroid. Example 2 - Graphs Take a graph G, with edges E. Let I be all the subsets of E without any cycles (circuits), ie. the subforests of G. The M = (E, I) is a matroid, known as the graphic matroid. A matroid can be defined in many ways. The vector space analogy adopted earlier is just one way.

3. Some more notation Useful terms: Given a matriod M = (E, I), a subset of E that is not in I is called dependent. A minimal dependent set is known as a circuit. A basis is a maximal independent set, ie. the the maximal feasible set. The collection of bases is denoted B(M) or just B. 1 1 3 has a circuit 2 2 4 3 5

4. Some more notation 1. For vectors, a basis is a set of linearly independent vectorsthat span E, such as {(0,0,1),(0,1,0),(1,0,0)} is a basis for 3, for example. 2. For graphs, a basis is a spanning tree. The definition of a matroid given above is just one way of describing a matroid, and there are other ways: Let E be a set, I be a collection of subsets of E. Then I is a collection of independent sets of a matroid M=(E,I) iff I satisfies axioms 1,2 as given before, and 3' as follows: 3') If A is any subset of E, then all maximal subsets X of A with X in I have the same cardinality.