This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossingfree drawings with vertices located at the grid points of phi? We characterize the trees that can be drawn on a two dimensional c * n × k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal -- it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.

We provide O(n)-time algorithms for constructing the following types of drawings of n-vertex 3-connected planar graphs: ffl 2D convex grid drawings with (3n) × (3n/2) area under the edge L 1 -resolution rule; ffl 2D strictly convex grid drawings with O(n³) × O(n³) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) × O(n) area under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers; ffl 3D convex drawings with O(1) × O(1) × O(n) volume under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers. We also

this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.

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.

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We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of non-planar graphs with few slopes are also considered. For example, interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree.

Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an area-optimal 2-visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)-time query support. All algorithms in this paper run in linear time.

We provide O(n)-time algorithms for constructing the following types of drawings of n-vertex 3-connected planar graphs: ffl 2D convex grid drawings with (3n) \Theta (3n=2) area under the edge L1 -resolution rule; ffl 2D strictly convex grid drawings with O(n 3 ) \Theta O(n 3 ) area under the edge resolution rule; ffl 2D strictly convex drawings with O(1) \Theta O(n) area under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers; ffl 3D convex drawings with O(1) \Theta O(1) \Theta O(n) volume under the vertex-resolution rule, and with vertex coordinates represented by O(n log n)-bit rational numbers. We also show the following lower bounds: ffl For infinitely many n-vertex graphs G, if G has a straightline 2D convex drawing in a w \Theta h grid satisfying the edge L1 -resolution rule then w;h 5n=6 +\Omega\Gamma20 and w + h 8n=3 + \Omega\Gamma838 ffl For infinitely many bounded-degree triconnected planar graphs G with n ver...

In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...

Abstract As cyberspace becomes an integral part of our daily life, its mastering becomes harder. To help, cyberspace can be represented by resources arranged in a multidimensional space. With geographical maps to exhibit the topology of this virtual space, people can have a better visual understanding. In this paper, methods focusing on the construction of lower dimension representations of this space are examined and illustrated with the World-Wide Web. It is expected that this work will contribute to addressing issues of navigation in cyberspace and, especially, avoiding the lost-in-cyberspace syndrome. Résumé Alors que le cyberspace envahit notre vie quotidienne, sa maîtrise devient de plus en plus complexe. On peut l’imaginer comme un ensemble de ressources arrangées dans un espace multi-dimensionnel. En utilisant des cartes géographiques pour représente la topologie virtuelle de cet espace, on arrive à mieux le comprendre, le cerner. Dans ce papier, des méthodes se concentrant sur la construction de représentations à dimensions réduites sont étudiées en les appliquant au World-Wide Web. On espère que ce travail contribuera à résoudre les problèmes de navigation dans ce monde virtuel et en particulier à éviter de s’y perdre. Ubersicht In einer Zeit, in der der Cyberspace ein integraler Bestandteil unseres täglichen Lebens wird, wird seine Beherrschung zunehmend schwieriger. Zur Erleichterung kann Cyberspace anhand von Quellen, angeordnet in einem multidimensionalen Raum, dargestellt werden. Mit geographischen Karten, die die Topologie dieses künstlichen Raumes aufzeigen, kann das visuelle Verständnis verbessert werden. In dieser Arbeit werden Methoden zur Konstruktion von Darstellungen mit niedriger Dimension dieses Raumes untersucht und anhand des World-Wide Web verdeutlicht.