Pathwidth And Layered Drawings Of Trees

An h-layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end-vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper h-layer drawing contains only proper edges, a short h-layer drawing contains no long edges, an upright h-layer drawing contains no flat edges, and an unconstrained h-layer drawing contains any type of edge. We prove optimal upper and lower bounds for each type of layered drawing (proper, short, upright, unconstrained) and give linear-time algorithms for obtaining drawings matching each upper bound. We note that the optimality of the upper bound for unconstrained layered drawings contradicts Proposition 1 of [8], and the optimality of the upper bound for short layered drawings contradicts Theorem 2 also of [8].