Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Area-Universal Rectangular Layouts

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure

Orientation-Constrained Rectangular Layouts

We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada, August 2009. 12 pages, 5 figure

Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

Let $G$ be an $n$-node planar graph. In a visibility representation of $G$, each node of $G$ is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of $G$ are vertically visible to each other. In the present paper we give the best known compact visibility representation of $G$. Given a canonical ordering of the triangulated $G$, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated $G$ yields a visibility representation of $G$ no wider than $\frac{22n-40}{15}$. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether $\frac{3n-6}{2}$ is a worst-case lower bound on the required width. Also, if $G$ has no degree-three (respectively, degree-five) internal node, then our visibility representation for $G$ is no wider than $\frac{4n-9}{3}$ (respectively, $\frac{4n-7}{3}$). Moreover, if $G$ is four-connected, then our visibility representation for $G$ is no wider than $n-1$, matching the best known result of Kant and He. As a by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Berlin, Germany, 200

Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem

We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of rectangles intersects. First, we investigate combinatorial contact arrangements, i.e., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph. We show that such rectangle arrangements are in bijection with the 4-orientations of an underlying planar multigraph and prove that there is a corresponding geometric rectangle contact arrangement. Moreover, we prove that every triangle-free planar graph is the contact graph of such an arrangement. Secondly, we introduce the question whether a given rectangle arrangement has a combinatorially equivalent square arrangement. In addition to some necessary conditions and counterexamples, we show that rectangle arrangements pierced by a horizontal line are squarable under certain sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the International Symposium on Graph Drawing and Network Visualization (GD) 201