Centers of sets of pixels (Q1570843)

A pixel set is a subset of a plane rectangular regular grid, under the graph structure defined by 4-neighbor adjacency (at city-block distance 1). A pixel (point, vertex) is a center of the set when its maximum graph-distance to any other vertex is minimal. It is shown that the set of centers of a simply connected pixel set (i.e. the complementary pixel set is also connected) is either a \(2\times 2\) square block, a diagonal staircase or a main diagonal segment without gaps. For 8-neighbor adjacency (at chessboard distance 1) other shapes are possible, but no characterization is obtained.