IsConnected returns true if the input graph is a connected graph or false otherwise. If G is a directed graph then the directions of edges are ignored. Use the IsStronglyConnected command to test whether each pair of vertices is connected by a directed path.

ConnectedComponents returns the components of the graph as a list of lists of vertices.  Each sublist is a list of vertices for a connected component.

A graph G is connected if for each pair of vertices u and v in G there exists a path from u to v in G (if G is undirected), or in the underlying graph of G (if G is directed).

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$

$G≔\mathrm{CycleGraph}\left(4\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

$\mathrm{IsConnected}\left(G\right)$

$\mathrm{true}$

$H≔\mathrm{GraphComplement}\left(G\right)$

$H≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 2\; edge(s)}$

$\mathrm{IsConnected}\left(H\right)$

$\mathrm{false}$

$\mathrm{ConnectedComponents}\left(H\right)$

$\left[\left[1\,3\right]\,\left[2\,4\right]\right]$

$\mathrm{DrawGraph}\left(H\right)$

$G≔\mathrm{Graph}\left(\left[1\,2\,3\,4\,5\,6\right]\,\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{4\,5\right\}\right\}\right)$

$G≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 3\; edge(s)}$

$\mathrm{IsConnected}\left(G\right)$

$\mathrm{false}$

$\mathrm{ConnectedComponents}\left(G\right)$

$\left[\left[1\,2\,3\right]\,\left[4\,5\right]\,\left[6\right]\right]$

$G≔\mathrm{OctahedronGraph}\left(\right)$

$G≔\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 12\; edge(s)}$

$\mathrm{ConnectedComponents}\left(G\right)$

$\left[\left[1\,2\,3\,4\,5\,6\right]\right]$

$H≔\mathrm{GraphComplement}\left(G\right)$

$H≔\mathrm{Graph\; 5:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 3\; edge(s)}$

$\mathrm{IsConnected}\left(H\right)$

$\mathrm{false}$

$\mathrm{ConnectedComponents}\left(H\right)$

$\left[\left[1\,2\right]\,\left[3\,4\right]\,\left[5\,6\right]\right]$

$\mathrm{DrawGraph}\left(G\right)$

$\mathrm{DrawGraph}\left(H\right)$