are connected to each other by a path in ). This flat may be identified with the partition of the vertices of into the connected components of the subgraph formed by : Every set of edges having the same closure as gives rise to the same partition of the vertices, and 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 whenever the partition corresponding to flat is a refinement of the partition corresponding to flat . In this aspect of graphic matroids, the graphic matroid for a complete graph 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 is naturally isomorphic to the lattice of partitions of an -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 can be defined as the column matroid of any oriented incidence matrix of . Such a matrix has one row for each vertex, and one column for each edge. The column for edge has in the row for one endpoint, in the row for the other endpoint, and 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 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 . 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