there is a path from or equivalently a walk (a path allowing repeated vertices and edges). Reachability is an equivalence relation, since: It is reflexive: There is a trivial path of length zero from any vertex to itself. It is symmetric: If there is a path from the same edges in the reverse order form a path from It is transitive: If there is a path from and a path from the two paths may be concatenated together to form a walk from The equivalence classes of this relation partition the vertices of the graph into disjoint sets, subsets of vertices that are all reachable from each other, with no additional reachable pairs outside of any of these subsets. Each vertex belongs to exactly one equivalence class. The components are then the induced subgraphs formed by each of these equivalence classes. Alternatively, some sources define components as the sets of vertices rather than as the subgraphs they induce. Similar definitions involving equivalence classes have been used to defined components for other forms of graph connectivity, including the weak components and strongly connected components of directed graphs and the biconnected components of undirected graphs. Number of components The number of components of a given finite graph can be used to count the number of edges in its spanning forests: In a graph with vertices and components, every spanning forest will have exactly edges. This number is the matroid-theoretic rank of the graph, and the rank of its graphic matroid. The rank of the dual cographic matroid equals the circuit rank of the graph, the minimum number of edges that must be removed from the graph to break all its cycles. In a graph with edges, vertices and components, the circuit rank is A graph can be interpreted as a topological space in multiple ways, for instance by placing its vertices as points in general position in three-dimensional Euclidean space and representing its edges as line segments between those points. The components of a graph can be generalized through these interpretations as the topological connected components of the corresponding space; these are equivalence classes of points that cannot be separated by pairs of disjoint closed sets. Just as the number of connected components of a topological space is an important topological invariant, the zeroth Betti number, the number of components of a graph is an important graph invariant, and in topological graph