Tessellations derived from random geometric graphs

TL;DR

This paper introduces a new class of random tessellations derived from planar geometric graphs, allowing for complex cell topologies and generalizing classical random tessellation models.

Contribution

It develops a theoretical framework for tessellations based on random geometric graphs, extending existing models to include non-convex, possibly with holes, and topologically complex cells.

Findings

Generalized formulae for random tessellations with complex cell topology

Framework accommodating cells with holes and various valencies

Extension of classical tessellation theory to new graph-based structures

Abstract

In this paper we consider a random partition of the plane into cells, the partition being based on the nodes and links of a {\it random planar geometric graph}. The resulting structure generalises the \emph{random \tes}\ hitherto studied in the literature. The cells of our partition process, possibly with holes and not necessarily closed, have a fairly general topology summarised by a functional which is similar to the Euler characteristic. The functional can also be extended to certain cell-unions which can arise in applications. Vertices of all valencies, $0, 1, 2, ...$ are allowed. Many of the formulae from the traditional theory of random tessellations with convex cells, are made more general to suit this new structure. Some motivating examples of the structure are given.