The DeleteVertex command returns a copy of the graph G with the specified vertex, or list of vertices, deleted.  The graph G is not changed. The resulting graph is the subgraph of G induced by the remaining vertices. Any edges adjacent to V are removed.

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

$G≔\mathrm{CycleGraph}\left(5\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$H≔\mathrm{DeleteVertex}\left(G\,\left[1\,4\right]\right)$

$H≔\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 3\; vertices\; and\; 1\; edge(s)}$

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

$\left\{\left\{2\,3\right\}\right\}$

$J≔\mathrm{InducedSubgraph}\left(G\,\left\{2\,3\,5\right\}\right)$

$J≔\mathrm{Graph\; 3:\; an\; undirected\; unweighted\; graph\; with\; 3\; vertices\; and\; 1\; edge(s)}$

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

$\left\{\left\{2\,3\right\}\right\}$

$\mathrm{DeleteVertex}\left(G\,6\right)$

Error, (in GraphTheory:-DeleteVertex) second argument expected to be a vertex or a list of vertices of the graph