The AddArc command adds one or more arcs to a directed graph. By default, the original digraph is changed to a digraph containing the specified set of arc(s).  By setting inplace=false, the original digraph remains unchanged and a new digraph containing the specified set of arcs is created.

If the graph is weighted, then a weighted arc can be added by calling AddArc with one or more arcs in the form $\left[\mathrm{arc}\,\mathrm{weight}\right]$, where the arc is just the from/to vertex pair, and the weight represents the value of the arc weight.

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

$G≔\mathrm{Digraph}\left(\left[a\,b\,c\,d\,e\right]\,\left\{\left[a\,b\right]\,\left[b\,c\right]\,\left[c\,d\right]\,\left[d\,e\right]\right\}\right)$

$G≔\mathrm{Graph\; 1:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 4\; arc(s)}$

$\mathrm{AddArc}\left(G\,\left[a\,c\right]\,\mathrm{inplace}=\mathrm{false}\right)$

$\mathrm{Graph\; 2:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 5\; arc(s)}$

$G$

$\mathrm{Graph\; 1:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 4\; arc(s)}$

$\mathrm{AddArc}\left(G\,\left\{\left[a\,c\right]\,\left[b\,d\right]\right\}\,\mathrm{inplace}=\mathrm{false}\right)$

$\mathrm{Graph\; 3:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 6\; arc(s)}$

$\mathrm{AddArc}\left(G\,\left\{\left[a\,c\right]\,\left[b\,d\right]\right\}\right)$

$\mathrm{Graph\; 1:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 6\; arc(s)}$

$G$

$\mathrm{Graph\; 1:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 6\; arc(s)}$

$\mathrm{Gw}≔\mathrm{Graph}\left(\mathrm{Matrix}\left(\left[\left[0\,1\,1\,0\right]\,\left[1\,0\,0\,3\right]\,\mathrm{`\$`}\left(\left[\mathrm{`\$`}\left(0\,4\right)\right]\,2\right)\right]\right)\,'\mathrm{weighted}'\,'\mathrm{directed}'\right)$

$\mathrm{Gw}≔\mathrm{Graph\; 4:\; a\; directed\; weighted\; graph\; with\; 4\; vertices\; and\; 4\; arc(s)}$

$\mathrm{Edges}\left(\mathrm{Gw}\,\mathrm{weights}\right)$

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

$\mathrm{AddArc}\left(\mathrm{Gw}\,\left[\left[1\,4\right]\,2\right]\right)$

$\mathrm{Graph\; 4:\; a\; directed\; weighted\; graph\; with\; 4\; vertices\; and\; 5\; arc(s)}$

$\mathrm{Edges}\left(\mathrm{Gw}\,\mathrm{weights}\right)$

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

$G≔\mathrm{Digraph}\left(\left[1\,3\,5\,7\right]\right)$

$G≔\mathrm{Graph\; 5:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 0\; arc(s)}$

$\mathrm{AddArc}\left(G\,\mathrm{Trail}\left(1\,5\,3\,7\,1\right)\right)$

$\mathrm{Graph\; 5:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 4\; arc(s)}$

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

$\left\{\left[1\,5\right]\,\left[3\,7\right]\,\left[5\,3\right]\,\left[7\,1\right]\right\}$