Area-Universal Rectangular Layouts

• Area-universal rectangular layouts are partitions of an axis-aligned rectangle into subrectangles that can realize any set of positive area assignments while maintaining fixed adjacencies.
• They are defined by a one-sidedness condition on maximal segments, which ensures efficient, polynomial-time recognition and unique area realization.
• These layouts have practical applications in cartograms, VLSI floorplanning, and treemap visualizations, where precise area allotment is critical.

An area-universal rectangular layout is a partition of an axis-aligned rectangle into a finite set of smaller, interior-disjoint rectangles such that, for any assignment of positive real areas to the rectangles, there exists a combinatorially equivalent layout that realizes the same adjacencies and prescribed areas. This concept is fundamental in rectangular cartograms, VLSI floorplanning, and spatial treemap layouts, where the ability to instantiate arbitrary area weights—while preserving a fixed contact graph structure—is essential. The class of area-universal layouts is characterized by a simple, robust combinatorial criterion, admits efficient algorithms for recognition and construction, and has deep connections to planar graph theory, distributive lattices, and combinatorial geometry.

1. Formal Definition and Combinatorial Characterization

A rectangular layout $L$ is defined as a partition of a closed bounding rectangle into finitely many closed subrectangles $R_1, \dots, R_n$ with pairwise disjoint interiors and whose union is the bounding rectangle. A crucial requirement is that no four rectangles meet at a single point, ensuring a contact system conducive to planar duality.

The dual graph $G(L)$ of a layout $L$ is the planar graph with one vertex for each rectangle and an edge between two vertices if their associated rectangles share a nondegenerate boundary segment. Two layouts $L, L'$ with $G(L) = G(L')$ are combinatorially equivalent if they induce identical regular edge labelings (as in Kant–He 1997), preserving the cyclic order and nature (horizontal/vertical) of adjacencies at each vertex.

A layout $L$ is area-universal if, for every assignment of positive real numbers $w(R_i)$, there is a combinatorially equivalent layout $L' \sim L$ in which each rectangle $R_i$ has area $w(R_i)$ (0901.3924).

The structural underpinning of area-universality is the property of one-sidedness of maximal segments. A maximal segment in $L$ is a maximal connected straight boundary arc formed by concatenating interior edges. $L$ is one-sided if every maximal segment forms the entire side of at least one rectangle, and all T-junctions along the segment point in the same direction (away from that rectangle).

Theorem (Eppstein et al. 2012): A rectangular layout is area-universal if and only if it is one-sided; that is, every maximal segment is the complete side of some rectangle (0901.3924).

2. Algorithmic Aspects and Efficient Recognition

Testing whether a given rectangular layout is area-universal reduces to checking the one-sidedness condition for each maximal interior segment. This can be achieved in $O(n)$ time for a layout with $n$ rectangles: enumerate all maximal segments, determine the incident rectangles on both sides, and verify that for each segment, all but one side contains at most a single rectangle. All T-junctions along the segment are inspected for coherent orientation.

Given a planar triangulation (proper dual graph), area-universal rectangular layouts can be constructed by decomposing the dual graph into components on separating 4-cycles, building the partial order of rectangular duals (as distributive lattices using Fusy’s flip lattice), and searching for one-sided layouts using parameterized or brute-force enumeration. The overall complexity is polynomial in fixed parameter $K$ (degree-4 flips per component) (0901.3924). For classes such as the special class $\mathcal{C}$ of rectangularly dualizable graphs specified by growth from a seed path of degree-4 vertices, explicit polynomial or near-linear time algorithms are available (Kumar et al., 2021).

For general planar graphs, recent advances guarantee a polynomial-time decision and construction algorithm for area-universal layouts via the slant–REL approach: The layout's regular edge labeling must be "slant" (each face in both colored orientation graphs has at least one side of length two), and candidate v-chains are pruned via a hierarchical DAG structure to ensure conflict-free realization (Wang, 2013).

3. Uniqueness and Realization

For any combinatorial layout $L$ and any strictly positive area vector $w$, there exists at most one realization $L' \sim L$ (up to independent horizontal and vertical scaling) such that the area of each rectangle matches its prescribed value (0901.3924). The proof is based on the "push graph": After superimposing two candidate realizations, the difference in coordinates forms a directed graph where acyclicity ensures that only one solution, modulo scaling, is possible.

For perimeter assignments (prescribing per-rectangle perimeter), uniqueness up to scaling holds, but existence fails for arbitrary assignments unless specific linear and feasibility conditions are met; these can be decided and realized in polynomial time (Gaussian elimination plus 2D linear programming) (0901.3924).

Realizing the areas from a one-sided layout (either rectangular or low-complexity rectilinear) is typically reduced to solving a sparse system of quadratic equations—each corresponding to the area constraint for a rectangle and adjacency constraints for maximal segments. Numerical methods (e.g., Newton iteration or hill-climbing pressure heuristics) converge rapidly for practical instances (Alam et al., 2011).

4. Necessary and Sufficient Conditions for Dual Graphs

The existence of an area-universal rectangular layout realizing a given planar graph as its dual fundamentally depends on the combinatorics of the graph and the chosen embedding.

For outerplanar graphs, a complete characterization holds: A biconnected outerplanar triangulated graph $G$ admits an area-universal rectangular dual if and only if it has exactly two degree-2 vertices. The construction proceeds by augmenting $G$ with four cardinal vertices, then choosing adjacencies so that the resulting extended graph and its regular edge labeling are guaranteed to be one-sided (and hence area-universal). The entire process, including the solution for arbitrary area vectors, is algorithmically linear in $n$ (Suthar et al., 20 Jan 2026).

In the broader context of rectangularly dualizable graphs, certain hereditary subclasses such as $\mathcal{C}$—those capable of being grown from a seed path of degree-4 vertices while at each stage exposing at most three new neighbors—admit efficient polynomial-time construction of area-universal layouts (Kumar et al., 2021). The extension to the full class of proper triangulated planar graphs is facilitated through slant–REL algorithms and backtracking-based conflict resolution (Wang, 2013).

5. Applications and Polygonal Complexity

Area-universal rectangular layouts are central in weighted cartogram construction, where regions (rectangles or simple polygonal contacts) represent, for instance, states or districts with assigned data such as population or GDP. By using area-universal layouts, one enables the simultaneous display of multiple area-weighted data sets without changing adjacencies or topological structure (0901.3924, Alam et al., 2011).

Applications include:

• Rectangular cartograms in geographic visualization.
• VLSI design and architectural floorplanning for reconfigurable modules (0901.3924).
• Treemaps and hierarchical data visualization for aspect ratio and adjacency control.

Polygonal complexity is a vital metric. For maximal planar graphs, every area-universal rectilinear dual can be realized with polygons of 8 sides—this bound is tight, with explicit constructions and lower bound examples (Alam et al., 2011). For Hamiltonian maximal planar graphs, the complexity remains 8, but further reductions to 6 sides are possible in the presence of one-legged Hamiltonian cycles, as in maximal outerplanar graphs.

6. Recent Developments and Open Problems

Recent progress includes the following:

• Full polynomial-time algorithms for deciding and constructing area-universal rectangular layouts for arbitrary proper planar triangulations via slant–RELs, including rigorous detection and elimination of combinatorially conflicting candidate chains (Wang, 2013).
• Complete structural characterization and explicit linear-time construction for outerplanar graphs, hinging on the degree-2 condition (Suthar et al., 20 Jan 2026).
• The introduction of efficient constructions for natural subclasses of rectangularly dualizable graphs and the formalization of the growth condition underlying area-universality in these classes (Kumar et al., 2021).

Outstanding questions include:

• Extension of efficient recognition and construction algorithms to broader or more general classes of planar graphs, possibly via further decompositions or combinatorial insights (e.g., beyond the class $\mathcal{C}$).
• Combinatorial characterization of area-universal rectilinear layouts (not strictly rectangular) and the search for minimal polygonal complexity in weighted cases (Alam et al., 2011).
• Refined algorithms for the direct, combinatorial realization of area assignments—avoiding floating-point iteration—in large-scale applications.

7. Illustrative Example and Tabular Summary

Consider a simple four-region layout: A single rectangle partitioned by a central vertical and central horizontal segment (yielding a “plus” pattern). This layout is one-sided, as each maximal interior segment forms the entire side of one rectangle, and hence is area-universal. Any assignment of four positive areas can be uniquely realized by choosing the positions of the two segments such that the corresponding areas match.

Research Area	Key Structural Condition	Algorithmic Status
Rectangular layouts (general)	One-sidedness of all maximal segments	Polynomial time for fixed $K$ (0901.3924, Wang, 2013)
Outerplanar graphs	Exactly two degree-2 vertices	Linear time (Suthar et al., 20 Jan 2026)
Class $\mathcal{C}$ RDGs	Growth from seed path, ≤3 new neighbors	Near-linear time (Kumar et al., 2021)
Maximal planar graphs	Existence via flip lattices and slant–RELs	Polynomial time (Wang, 2013)

The theory of area-universal rectangular layouts thus provides a robust combinatorial and algorithmic foundation for the construction of adaptable, adjacency-preserving spatial representations with prescribed area data, with proven optimality and efficiency for a range of important graph classes and applications (0901.3924, Wang, 2013, Kumar et al., 2021, Suthar et al., 20 Jan 2026, Alam et al., 2011).