Planar lattice subsets with minimal vertex boundary

TL;DR

This paper provides a complete geometric characterization of minimal vertex boundary sets in the planar integer lattice, revealing their structure, classification, and connectivity properties.

Contribution

It introduces a geometric characterization of minimal sets in the planar lattice and explores their structure, classification, and the properties of the graph formed by these sets.

Findings

Characterized all minimal sets of the planar lattice geometrically.

Classified uniquely minimal and efficient sets within the lattice.

Analyzed the structure and properties of the graph of minimal sets, including connectivity and components.

Abstract

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We characterize uniquely minimal sets of X: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of X: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph G of minimal sets whose vertices are congruence classes of minimal sets of X and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.