Rectilinear Duals Using Monotone Staircase Polygons

Abstract

A rectilinear dual of a plane graph refers to a partition of a rectangular area into nonoverlapping rectilinear polygonal modules, where each module corresponds to a vertex such that two modules have side-contact iff their corresponding vertices are adjacent. It is known that 8-sided rectilinear polygons are sufficient and necessary to construct rectilinear duals of maximal plane graphs. The result stands even if modules are restricted to T-shape polygons. We show that the optimum polygonal complexity of T-free rectilinear duals is exactly 12. It justifies the intuition that T-shape is more powerful than other 8-sided modules. Our construction of 12-sided T-free rectilinear duals only requires monotone staircase modules. We also consider the issue of area-universality, and show that monotone staircase modules are not sufficient to construct area-universal rectilinear duals in general even when an unbounded polygonal complexity is allowed; however, eight sides are sufficient for Hamiltonian plane graphs. This line of research regarding monotone staircase modules is also motivated by the so-called monotone staircase cuts in VLSI floorplanning. We feel that our results provide a new insight towards a comprehensive understanding of modules in rectilinear duals.