Counting Triangulations of Fixed Cardinal Degrees

TL;DR

The paper proves that counting PSL triangulations compatible with fixed per-vertex cardinal-direction degrees is #P-hard. It does so by a gadget-based reduction that encodes noncrossing tile selections within a tiling into a cardinal-signature realization, then shows a bijection between these selections and realizations in a carefully constructed class $\mathcal{R}$ containing a frame graph and gadget realizations. The reduction hinges on introducing an auxiliary problem (#tiled noncrossing cycle-set) and linking it to #3-regular bipartite planar vertex cover, enabling a parsimonious chain to #cardinal signature realization. The work also shows the hardness extends to degree information in any $d\ge4$ directions via shear and discusses implications for inverse problems in directional transforms of topological data analysis, as well as several open questions about existence decisions for PSL realizations.

Abstract

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if we know, for each vertex, the number of neighbors in each of the four cardinal directions, the triangulation is not completely determined. In fact, we show that counting such triangulations is #P-hard via a reduction from #3-regular bipartite planar vertex cover.

Figures (11)

• Figure 1: (Left) A pair of maximal PSL triangulations with identical vertices and north- and south-degrees. (Right) A pair with identical cardinal signatures. Common edges are shown in black.
• Figure 2: A graph whose signature saturates the halfspace below the height of $v$, thereby forcing all edges in the boundary of the convex hull of this lower subgraph.
• Figure 3: The seventeen different tile types in the #tiled noncrossing cycle-set problem. In any tiling, the four tile types in (h) always appear together as illustrated. In contrast to (a--f), which each have both a red and a blue tile type, the four tile types in (h) do not need a counterpart with colors exchanged.
• Figure 4: (a) An instance of #3-regular bipartite planar independent set. (b) A corresponding set of cycles in the plane. (c) Part of a corresponding tiling.
• Figure 5: (Left) A frame cell, which can be sheared to be arbitrarily close to square while retaining the specification of Construction \ref{['cons:framecell']}. (Right) A $3\times 3$ frame graph, as in Construction \ref{['cons:framegraph']}. The regions that are forbidden from containing vertices in their interiors are indicated by shaded orange.

Theorems & Definitions (30)

• Definition 2: Cardinal Degrees
• Definition 3: Forced Edges
• Lemma 4
• Corollary 5
• Lemma 6
• Lemma 6
• Lemma 7: Forced Frame Edges
• Definition 8: Gadget
• Lemma 10
• Lemma 11