An O(V) algorithm M for computing transitive reduction of a planar acyclic digraph

We present a linear O(V) algorithm for computing the transitive reduction of a planar acyclic digraph G, where V is the number of nodes in G. The algorithm makes explicit use of a fixed, but otherwise arbitrary, planar representation of G and obtains the transitive reduction in two steps, by computing successively the left reduction and the right-reduction. The planar digraphs form the second class of digraphs for which linear transitive reduction algorithm is known; the other class being the digraphs whose transitive reductions are directed spanning trees.