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 show that the neighborhood of a minimal set is minimal. 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.
Highlights
The classical isoperimetric problem can be stated as follows: amongst all closed curves in the plane with fixed length, characterize those that enclose the maximal area
In this article we study the graph X = Z21 with vertex set X0 = Z2 and edges connecting all pairs of vertices 1-distance one apart
Much of the literature has focused on exhibiting a sequence of minimal sets, i.e. a sequence (An) where each An is a minimal set consisting of exactly n vertices
Summary
The classical isoperimetric problem can be stated as follows: amongst all closed curves in the plane with fixed length, characterize those that enclose the maximal area. While circles are the natural geometric solution to Problem 1, there can be many different congruence classes of minimal sets in X of a given size and our result exactly describes these solutions. This approach lets us prove many applications that allow us to better understand the collection of all minimal sets. Our main theorem, stated below, gives two related characterizations of minimal sets in terms of their enclosing boxes. We note that by Theorem E we get an explicit characterization of mortal and dead sets in terms of box parametrizations. A minimal set in X is connected if and only if it is not congruent to B(0, 2)
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