Connected Component

A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if  and  then . Hence, being in the same component is an equivalence relation, and the equivalence classes are the connected components.

Using pathwise-connectedness, the pathwise-connected component containing  is the set of all  pathwise-connected to . That is, it is the set of  such that there is a continuous path from  to .

Technically speaking, in some topological spaces, pathwise-connected is not the same as connected. A subset  of  is connected if there is no way to write  with  and  disjoint open sets. Every topological space decomposes into a disjoint union  where the  are connected. The  are called the connected components of .

The connected components of a graph  are the set of largest subgraphs of  that are each connected. Connected components of a graph may be computed in the Wolfram Language as ConnectedComponents[g] (returned as lists of vertex indices) or ConnectedGraphComponents[g] (returned as a list of graphs). Precomputed values for a number of graphs are available as GraphData[g, "ConnectedComponents"].