Area-Universal and Constrained Rectangular Layouts

Abstract

Abstract.A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. These layouts are used 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 is desirable for one rectangular layout to 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. 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. Furthermore we show how to impose restrictions on the orientations of edges and junctions of the rectangular 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.

Stable Treemaps via Local Moves

“…Vertical maximal segments similarly define a partial order from left to right. Following Eppstein et al [7], we say that two layouts L and L are order-equivalent if the partial orders for L and L are isomorphic. 2 An example of order-equivalent layouts is given in Figure 3.…”

Proportional Contact Representations of 4-Connected Planar Graphs

“…However, the same cannot be said about proportional contact representations; there are instances of 4-connected planar graphs with triangular internal faces and a non-triangular outerface that have no proportional contact representations with rectangles for some weight functions. Recently, Eppstein et al [5] characterized the class of area-universal rectangular duals, i.e., rectangular duals that can realize any specified area for the rectangles. In summary, the best known lower and upper bounds for hole-free proportional contact representations of 4-connected planar graphs are 4 and 8, respectively [2,4].…”

“…A line-segment not contained in any other line-segment is maximal. A maximal line-segment s is called one-sided if it forms a full side of at least one rectangular face, or in other words, if the proved that if all the maximal segments in a rectangular dual Γ are one-sided then Γ is area-universal, which means that any distribution of areas to the rectangles in Γ can be realized with a topologically equivalent layout [5]. Unfortunately, not every internally triangulated 4-connected plane graph has a rectangular dual that is also area-universal.…”

“…Proof Sketch. If Γ contains no two-sided maximal segment, then it can realize any weight function [5] and we are done. Otherwise, for each horizontal two-sided maximal segment s, we replace s by a rectangle with a small height (< the minimum feature size of Γ ) and with horizontal span same as s; see Fig.…”

On Contact Graphs with Cubes and Proportional Boxes

“…Representations with other axis-aligned and non-axis-aligned polygons [7,11,19] have been studied. Related graph-theoretic, combinatorial and geometric problems continue to be of interest [6,8,12]. The weighted variant of the problem has been considered in the context of rectangular, rectilinear, and unrestricted cartograms [4,9,15].…”