Unfolding \(3\)-separated polycube graphs of arbitrary genus (Q2096388)

An \textit{unfolding} of a polyhedron \(P\) is an isometric mapping from a non-self-intersecting plane polygon \(N\) (a ``net'' for \(P\)) to the boundary of \(P\), onto and 1-1 except for the boundary of \(N\). The best-known example is the Latin cross net for a cube. If (as in that case) the boundary of \(N\) maps into the 1-skeleton of \(P\), we call the unfolding an \textit{edge unfolding}. It is known that every convex polyhedron has an unfolding, but that a nonconvex polyhedron may have no edge unfolding. Whether all convex polyhedra have edge unfoldings, or all nonconvex polyhedra have unfoldings, are open questions. The authors define a \textit{3-separated polycube graph} as a union of unit cubes, glued along shared faces, such that no four cubes surround an edge and no two adjacent cubes each have more than two neighbours. Such a polyhedron need not be convex or even simply connected. The authors show that if we replace each face of a 3-separated polycube graph by a \(7\times 7\) grid of congruent squares, the resulting structure has an edge unfolding, which is an unfolding of the original.