Generalized Tilings with Height Functions

TL;DR

This paper introduces a broad generalization of tilings with height functions, enabling polynomial-time tilability decision, analysis of flip-accessibility graphs, and revealing lattice structures, with applications to known tiling problems.

Contribution

It presents a new generalized framework for tilings with height functions, allowing for polynomial-time tilability checks and structural analysis of flip graphs.

Findings

Tilability can be decided in polynomial time.

The flip-accessibility graph's connected components can be determined.

The directed flip graph forms a distributive lattice.

Abstract

In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be defined (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the undirected flip-accessibility graph can be determined, and the directed flip-accessibility graph induces a distributive lattice structure. Finally, we give a few examples of known tiling problems which can be viewed as particular cases of the new notions we introduce.