The InducedSubgraph command returns the subgraph induced by a specified set (or list) of vertices.

To construct a subgraph using a set of edges, use the Subgraph command.

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

$\mathrm{C6}≔\mathrm{CycleGraph}\left(6\right)$

$\mathrm{C6}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 6\; edges}$

$H≔\mathrm{InducedSubgraph}\left(\mathrm{C6}\,\left[1\,2\,6\right]\right)$

$H≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 3\; vertices\; and\; 2\; edges}$

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

$\left[1\,2\,6\right]$

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

$\left\{\left\{1\,2\right\}\,\left\{1\,6\right\}\right\}$

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

$P≔\mathrm{PetersenGraph}\left(\right)$

$P≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 10\; vertices\; and\; 15\; edges}$

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

$H≔\mathrm{InducedSubgraph}\left(P\,\left[1\,2\,3\,6\,7\,9\right]\right)$

$H≔\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 6\; edges}$

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