# Matroid

2010 Mathematics Subject Classification: Primary: 05B [MSN][ZBL]

A combinatorial abstraction of a linear algebra. A matroid is specified by a set $V$ of elements and a family $\mathcal{E} = \{E_1,E_2,\ldots \}$ of subsets of $V$, called independent subsets, which satisfy the following axioms: 1) the empty set is independent; 2) any subset of an independent set is independent; 3) for every subset $A \subseteq V$, all the independent subsets of the matroid which are contained in $A$ and are maximal with respect to inclusion relative to $A$ have the same number of elements.

Examples. 1) The set $V$ of rows of an arbitrary rectangular matrix and the family $\mathcal{E}$ of all subsets of $V$ consisting of linearly independent rows form a matroid. 2) Let $\mathcal{L} = \{L_1,L_2,\ldots\}$ be the set of all skeleton forests (see Tree) of a graph $G$, and let $R(L_i)$ be the set of edges of the forest $L_i$, $i=1,2,\ldots$. Then the set $V$ of edges of the graph $G$ and the family $\mathcal{E} = \{R(L_i) : i=1,2,\ldots\}$ form a matroid. 3) Let $G$ be a bipartite graph (cf. Graph, bipartite) with parts $W', W''$. A subset $V \subseteq W'$ of vertices for which there is a matching $P$ of the graph $G$ such that every vertex $v \in V$ is incident to some edge of $P$ is called a partial transversal. The set $W'$ and the set of all transversals of the graph $G$ form a so-called transversal matroid.

A matroid can be also specified by a set $V$ of elements and a family $\mathcal{C} = \{C_1,C_2,\ldots\}$ of non-empty subsets $C_i \subseteq V$, called circuits, which satisfy the following axioms: no proper subset of a circuit is a circuit; and if $v \in C_i \cap C_j$, then $C_i \cup C_j \setminus \{ v \}$ contains a circuit. The independent subsets of this matroid are the subsets $E \subseteq V$ that do not contain circuits.

If $G$ is a graph, then the set of its edges and the family of its simple circuits form what is called a graphic matroid. If for the circuits of a matroid one takes the cocycles (cuts, cf. Graph, connectivity of a) of the graph $G$, the resulting matroid is called a cographic matroid. The matroids of the last two types are also termed cyclic and cocyclic. The notion of a "matroid" is used in graph theory and combinatorics in the proof of some assertions on covering and packing of matchings.

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How to Cite This Entry:
Matroid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matroid&oldid=49841
This article was adapted from an original article by A.A. Sapozhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article