Efficient enumeration of transversal edge-partitions

An irreducible triangulation is a plane graph such that its outer face is a quadrangle, every inner face is a triangle, and it has no separating triangle. Let $T$ be an irreducible triangulation with $n$ vertices. A rectangular dual $R$ of $T$ is a dissection of a rectangle into (small) rectangles such that (1) each rectangle of $R$ corresponds to a vertex of $T$, and (2) two rectangles of $R$ are adjacent if the two corresponding vertices of $T$ are adjacent. Finding a rectangular dual of a given graph has an application on cartograms and VLSI floor-planning. In this paper, we consider the problem of enumerating all the rectangular duals of a given irreducible triangulation. It is known that the set of rectangular duals of an irreducible triangulation $T$ one-to-one corresponds to the set of transversal edge-partitions of $T$. Hence, in this paper, we design an enumeration algorithm of all the transversal edge-partitions of an irreducible triangulation with $n$ vertices. The proposed algorithm enumerates them in $O\left(n\right)$-delay and $O\left(n^2\right)$-space after $O(nlogn)$-time preprocessing.