Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)-area planar upward grid drawing that preserves the left-to-right ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...

A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.

A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as non-intersecting straight line segments. In this paper, we show that, if an internally triangulated plane graph G has no non-empty triangles (a non-empty triangle is a triangle of G containing some vertices in its interior), then G can be embedded on a grid of size W \Theta H such that W + H n, W (n + 3)=2 and H 2(n \Gamma 1)=3, where n is the number of vertices of G. Such an embedding can be computed in linear time. 1 Introduction Let G = (V; E) be a graph with n vertices. We always assume n 3 in this paper. G is planar if it can be drawn on the plane such that the vertices are located at distinct points, and the edges are represented by nonintersecting curves joining their endpoints. A plane graph is a planar graph with a fixed plane embedding. A straight line grid embedding of a planar graph is a drawing where the vertices are located at...