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weighted=true, false, or FAIL

Specifies whether the resulting graph should have edge weights corresponding to the weighted length of the shortest path in the input graph. The default is FAIL, meaning the graph is weighted if and only if the input graph is weighted.

The TransitiveClosure( G ) command constructs the graph which is the transitive closure of the graph G with respect to the edge relation.

The transitive closure of an undirected graph G is an undirected graph with the same vertices as G in which there is an edge between distinct vertices u and v whenever there is a path between u and v in G.

Similarly, the transitive closure of a directed graph G is a directed graph with same vertices as G, in which there is an arc from vertex u to vertex v whenever there is a directed path from u to v in G.

The transitive closure of a connected undirected graph will always be the complete graph on the vertices of the input graph.

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

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

$G≔\mathrm{Graph\; 1:\; a\; directed\; graph\; with\; 6\; vertices\; and\; 4\; arcs}$

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

$H≔\mathrm{Graph\; 2:\; a\; directed\; graph\; with\; 6\; vertices\; and\; 8\; arcs}$

$\mathrm{DrawGraph}\left(\left[G\,H\right]\,\mathrm{style}=\mathrm{circle}\right)$

$\mathrm{TCW}≔\mathrm{TransitiveClosure}\left(G\,\mathrm{weighted}=\mathrm{true}\right)$

$\mathrm{TCW}≔\mathrm{Graph\; 3:\; a\; directed\; weighted\; graph\; with\; 6\; vertices\; and\; 8\; arcs}$

$\mathrm{DrawGraph}\left(\mathrm{TCW}\,\mathrm{showweights}\,\mathrm{style}=\mathrm{circle}\right)$