Connected Component

A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if a∼b and b∼c then a∼c. 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 x in X is the set of all y pathwise-connected to x. That is, it is the set of y such that there is a continuous path from x to y.

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

The connected components of a graph G are the set of largest subgraphs of G 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"].

See also

Connected Set, Pathwise-Connected, s-Cluster, Strongly Connected Component, Topological Space, Weakly Connected Component

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Connected Component." From MathWorld--A Wolfram Web Resource.

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