A LEFT-FIRST SEARCH ALGORITHM FOR PLANAR GRAPHS

We give an O(\V(G)\)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacen t. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovic. The edges of any maximal bipartite plane graph G with outer face bwb'w' can be colored by two colors such that the c olor classes form spanning trees of G - b and G - b', respectively. Fu rthermore, such a coloring can be found in linear time. Our method is based on a new linear-time algorithm for constructing bipolar orientat ions of 2-connected plane graphs.