use Graph::TransitiveClosure;
    use Graph::Directed; # or Undirected

    my $g  = Graph::Directed->new;
    $g->add_...(); # build $g

    # Compute the transitive closure graph.
    my $tcg = Graph::TransitiveClosure->new($g);
    $tcg->is_reachable($u, $v) # Identical to $tcg->has_edge($u, $v)

    # Being reflexive is the default, meaning that null transitions
    # (transitions from a vertex to the same vertex) are included.
    my $tcg = Graph::TransitiveClosure->new($g, reflexive => 1);
    my $tcg = Graph::TransitiveClosure->new($g, reflexive => 0);

    # is_reachable(u, v) is always reflexive.
    $tcg->is_reachable($u, $v)

    # The reflexivity of is_transitive(u, v) depends of the reflexivity
    # of the transitive closure.
    $tcg->is_transitive($u, $v)

    # You can check any graph for transitivity.
    $g->is_transitive()

    my $tcg = Graph::TransitiveClosure->new($g, path_length => 1);
    $tcg->path_length($u, $v)

    # path_vertices is automatically always on so this is a no-op.
    my $tcg = Graph::TransitiveClosure->new($g, path_vertices => 1);
    $tcg->path_vertices($u, $v)

    # Both path_length and path_vertices.
    my $tcg = Graph::TransitiveClosure->new($g, path => 1);
    $tcg->path_vertices($u, $v)
    $tcg->length($u, $v)

    my $tcg = Graph::TransitiveClosure->new($g, attribute_name => 'length');
    $tcg->path_length($u, $v)

You can use Graph::TransitiveClosure to compute the transitive closure graph of a graph and optionally also the minimum paths (lengths and vertices) between vertices, and after that query the transitiveness between vertices by using the is_reachable() and is_transitive() methods, and the paths by using the path_length() and path_vertices() methods.

For further documentation, see the Graph::TransitiveClosure::Matrix.

new($g, %opt)

Construct a new transitive closure object. Note that strictly speaking the returned object is not a graph; it is a graph plus other stuff. But you should be able to use it as a graph plus a couple of methods inherited from the Graph::TransitiveClosure::Matrix class.

These are only the methods 'native' to the class: see Graph::TransitiveClosure::Matrix for more.

is_transitive($g)

Return true if the Graph $g is transitive.

transitive_closure_matrix

Return the transitive closure matrix of the transitive closure object.

The transitive closure matrix is stored as an attribute of the graph called _tcm, and any methods not found in the graph class are searched in the transitive closure matrix class.