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Title: Planar Lattice Subsets with Minimal Vertex Boundary
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.  more » « less
Award ID(s):
1840190
PAR ID:
10322929
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
The Electronic Journal of Combinatorics
Volume:
28
Issue:
3
ISSN:
1077-8926
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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