An Equivalence Relation
An alternative way to define connected components involves the equivalence classes of an equivalence relation that is defined on the vertices of the graph. In an undirected graph, a vertex v is reachable from a vertex u if there is a path from u to v. In this definition, a single vertex is counted as a path of length zero, and the same vertex may occur more than once within a path. 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 u to v, the same edges form a path from v to u.
- It is transitive: If there is a path from u to v and a path from v to w, the two paths may be concatenated together to form a path from u to w.
The connected components are then the induced subgraphs formed by the equivalence classes of this relation.
Read more about this topic: Connected Component (graph Theory)
Famous quotes containing the word relation:
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)