The Trail inert function is used as a short form description of edges in a graph passing through a vertex sequence/list in the given order. For example, Trail(1,2,3,4) or Trail([1,2,3,4]) are short forms to specify a trail through the vertices that generates the edges $\left[1\,2\right]$, $\left[2\,3\right]$ and $\left[3\,4\right]$.

The Trail function is only understood by the functions that construct graphs (Graph and Digraph) as well as functions that add edges to a graph or remove edges from a graph (AddEdge, DeleteEdge, AddArc, and DeleteArc). It is also used as a return value for a specified path (IsEulerian).

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

$G≔\mathrm{Graph}\left(4\,\mathrm{Trail}\left(1\,2\,3\,4\,1\right)\right)$

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

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

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

$G≔\mathrm{Digraph}\left(4\,\mathrm{Trail}\left(1\,2\,3\,4\,1\right)\right)$

$G≔\mathrm{Graph\; 2:\; a\; directed\; graph\; with\; 4\; vertices\; and\; 4\; arc(s)}$

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

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

$G≔\mathrm{Graph}\left(4\right)$

$G≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 0\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\mathrm{Trail}\left(1\,2\,3\,4\,1\right)\,'\mathrm{inplace}'=\mathrm{true}\right)$

$\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

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

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

$\mathrm{IsEulerian}\left(G\,'T'\right)$

$\mathrm{true}$

$T$

$\mathrm{Trail}\left(1\,2\,3\,4\,1\right)$

$G≔\mathrm{Digraph}\left(4\right)$

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

$\mathrm{AddArc}\left(G\,\mathrm{Trail}\left(4\,3\,2\,1\,4\right)\,'\mathrm{inplace}'=\mathrm{true}\right)$

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

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

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