Area-universality in Outerplanar Graphs

TL;DR

The work addresses the problem of characterizing when outerplanar graphs admit area-universal rectangular layouts and provides constructive methods to obtain them. It shows that a biconnected outerplanar graph has an area-universal layout iff it has exactly two degree-two vertices, linking this to the absence of flippable edges in any regular edge labeling via $r(\mathcal{G})$; when this condition holds, a linear-time augmentation procedure via Outer4Completion yields an extended outerplanar graph with area-universal layouts. For the degree-two case, the authors delineate when augmentations are unique and area-universal, and when multiple augmentations arise, enumerating all possible area-universal layouts realizing a given adjacency structure. The results provide a structural foundation and practical algorithms for constructing area-universal rectangular layouts, with implications for VLSI design and architectural floor planning, and they open avenues for broader generalizations beyond outerplanar graphs.

Abstract

A rectangular floorplan is a partition of a rectangle into smaller rectangles such that no four rectangles meet at a single point. Rectangular floorplans arise naturally in a variety of applications, including VLSI design, architectural layout, and cartography, where efficient and flexible spatial subdivisions are required. A central concept in this domain is that of area-universality: a floorplan (or more generally, a rectangular layout) is area-universal if, for any assignment of target areas to its constituent rectangles, there exists a combinatorially equivalent layout that realizes these areas. In this paper, we investigate the structural conditions under which an outerplanar graph admits an area-universal rectangular layout. We establish a necessary and sufficient condition for area-universality in this setting, thereby providing a complete characterization of admissible outerplanar graphs. Furthermore, we present an algorithmic construction that guarantees that the resulting layout is always area-universal.

Figures (12)

• Figure 1: (a) Presence of separating triangles $(2,4,9)$, $(6,9,10)$, and $(6,9,5)$. (b) A properly triangulated plane graph (PTPG). (c) An extended graph of a PTPG.
• Figure 2: (a) An extended outerplanar graph. (b) Presence of flippable edges $(5,7)$ and $(4,8)$. (c) Altered labels of the flippable edges $(5,7)$ and $(4,8)$.
• Figure 3: (a) A rectangular layout that is not area-universal. (b) An area-universal rectangular layout.
• Figure 4: The smallest outerplanar graph G not possessing any area-universal layout
• Figure 5: (a) Paths are separated by degree-2 vertices. (b-d) Three possible labeling patterns among $v_1, v_2,$ and $v_3$.

Theorems & Definitions (17)

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