Planar Lattice Subsets with Minimal Vertex Boundary

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