Matroid with graph forests as independent sets
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs.
Definition.
A matroid may be defined as a family of finite sets (called the "independent sets" of the matroid) that is closed under subsets and that satisfies the "exchange property": if sets formula_1 and formula_2 are both independent, and formula_1 is larger than formula_2, then there is an element formula_5 such that formula_6 remains independent. If formula_7 is an undirected graph, and formula_8 is the family of sets of edges that form forests in formula_7, then formula_8 is clearly closed under subsets (removing edges from a forest leaves another forest). It also satisfies the exchange property: if formula_1 and formula_2 are both forests, and formula_1 has more edges than formula_2, then it has fewer connected components, so by the pigeonhole principle there is a component formula_15 of formula_1 that contains vertices from two or more components of formula_2. Along any path in formula_15 from a vertex in one component of formula_2 to a vertex of another component, there must be an edge with endpoints in two components, and this edge may be added to formula_2 to produce a forest with more edges. Thus, formula_8 forms the independent sets of a matroid, called the graphic matroid of formula_7 or formula_23. More generally, a matroid is called graphic whenever it is isomorphic to the graphic matroid of a graph, regardless of whether its elements are themselves edges in a graph.
The bases of a graphic matroid formula_23 are the full spanning forests of formula_7, and the circuits of formula_23 are the simple cycles of formula_7. The rank in formula_23 of a set formula_29 of edges of a graph formula_7 is formula_31 where formula_32 is the number of vertices in the subgraph formed by the edges in formula_29 and formula_34 is the number of connected components of the same subgraph. The corank of the graphic matroid is known as the circuit rank or cyclomatic number.
The lattice of flats.
The closure formula_35 of a set formula_36 of edges in formula_23 is a flat consisting of the edges that are not independent of formula_36 (that is, the edges whose endpoints are connected to each other by a path in formula_36). This flat may be identified with the partition of the vertices of formula_7 into the connected components of the subgraph formed by formula_36: Every set of edges having the same closure as formula_36 gives rise to the same partition of the vertices, and formula_35 may be recovered from the partition of the vertices, as it consists of the edges whose endpoints both belong to the same set in the partition. In the lattice of flats of this matroid, there is an order relation formula_44 whenever the partition corresponding to flat formula_45 is a refinement of the partition corresponding to flat formula_46.
In this aspect of graphic matroids, the graphic matroid for a complete graph formula_47 is particularly important, because it allows every possible partition of the vertex set to be formed as the set of connected components of some subgraph. Thus, the lattice of flats of the graphic matroid of formula_47 is naturally isomorphic to the lattice of partitions of an formula_32-element set. Since the lattices of flats of matroids are exactly the geometric lattices, this implies that the lattice of partitions is also geometric. 
Representation.
The graphic matroid of a graph formula_7 can be defined as the column matroid of any oriented incidence matrix of formula_7. Such a matrix has one row for each vertex, and one column for each edge. The column for edge formula_52 has formula_53 in the row for one endpoint, formula_54 in the row for the other endpoint, and formula_55 elsewhere; the choice of which endpoint to give which sign is arbitrary. The column matroid of this matrix has as its independent sets the linearly independent subsets of columns.
If a set of edges contains a cycle, then the corresponding columns (multiplied by formula_54 if necessary to reorient the edges consistently around the cycle) sum to zero, and is not independent. Conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is independent. Therefore, the column matrix is isomorphic to formula_23.
This method of representing graphic matroids works regardless of the field over which the incidence is defined. Therefore, graphic matroids form a subset of the regular matroids, matroids that have representations over all possible fields.
The lattice of flats of a graphic matroid can also be realized as the lattice of a hyperplane arrangement, in fact as a subset of the braid arrangement, whose hyperplanes are the diagonals formula_58. Namely, if the vertices of formula_7 are formula_60 we include the hyperplane formula_61 whenever formula_62 is an edge of formula_7.
Matroid connectivity.
A matroid is said to be connected if it is not the direct sum of two smaller matroids; that is, it is connected if and only if there do not exist two disjoint subsets of elements such that the rank function of the matroid equals the sum of the ranks in these separate subsets. Graphic matroids are connected if and only if the underlying graph is both connected and 2-vertex-connected.
Minors and duality.
A matroid is graphic if and only if its minors do not include any of five forbidden minors: the uniform matroid formula_64, the Fano plane or its dual, or the duals of formula_65 and formula_66 defined from the complete graph formula_67 and the complete bipartite graph formula_68. The first three of these are the forbidden minors for the regular matroids, and the duals of formula_65 and formula_66 are regular but not graphic.
If a matroid is graphic, its dual (a "co-graphic matroid") cannot contain the duals of these five forbidden minors. Thus, the dual must also be regular, and cannot contain as minors the two graphic matroids formula_65 and formula_66.
Because of this characterization and Wagner's theorem characterizing the planar graphs as the graphs with no formula_67 or formula_68 graph minor, it follows that a graphic matroid formula_23 is co-graphic if and only if formula_7 is planar; this is Whitney's planarity criterion. If formula_7 is planar, the dual of formula_23 is the graphic matroid of the dual graph of formula_7. While formula_7 may have multiple dual graphs, their graphic matroids are all isomorphic.
Algorithms.
A minimum weight basis of a graphic matroid is a minimum spanning tree (or minimum spanning forest, if the underlying graph is disconnected). Algorithms for computing minimum spanning trees have been intensively studied; it is known how to solve the problem in linear randomized expected time in a comparison model of computation, or in linear time in a model of computation in which the edge weights are small integers and bitwise operations are allowed on their binary representations. The fastest known time bound that has been proven for a deterministic algorithm is slightly superlinear.
Several authors have investigated algorithms for testing whether a given matroid is graphic. For instance, an algorithm of solves this problem when the input is known to be a binary matroid. solves this problem for arbitrary matroids given access to the matroid only through an independence oracle, a subroutine that determines whether or not a given set is independent.
Related classes of matroids.
Some classes of matroid have been defined from well-known families of graphs, by phrasing a characterization of these graphs in terms that make sense more generally for matroids. These include the bipartite matroids, in which every circuit is even, and the Eulerian matroids, which can be partitioned into disjoint circuits. A graphic matroid is bipartite if and only if it comes from a bipartite graph and a graphic matroid is Eulerian if and only if it comes from an Eulerian graph. Within the graphic matroids (and more generally within the binary matroids) these two classes are dual: a graphic matroid is bipartite if and only if its dual matroid is Eulerian, and a graphic matroid is Eulerian if and only if its dual matroid is bipartite.
Graphic matroids are one-dimensional rigidity matroids, matroids describing the degrees of freedom of structures of rigid beams that can rotate freely at the vertices where they meet. In one dimension, such a structure has a number of degrees of freedom equal to its number of connected components (the number of vertices minus the matroid rank) and in higher dimensions the number of degrees of freedom of a "d"-dimensional structure with "n" vertices is "dn" minus the matroid rank. In two-dimensional rigidity matroids, the Laman graphs play the role that spanning trees play in graphic matroids, but the structure of rigidity matroids in dimensions greater than two is not well understood.
References.
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