Vertex-connectivity in periodic graphs and underlying nets of crystal structures

Periodic nets used to describe the combinatorial topology of crystal structures have been required to be 3-connected by some authors. A graph is n-connected when deletion of less than n vertices does not disconnect it. n-Connected graphs are a fortiari n-coordinated but the converse is not true. This article presents an analysis of vertex-connectivity in periodic graphs characterized through their labelled quotient graph (LQG) and applied to a definition of underlying nets of crystal structures. It is shown that LQGs of p-periodic graphs (p ≥ 2) that are 1-connected or 2-connected, but not 3-connected, are contractible in the sense that they display, respectively, singletons or pairs of vertices separating dangling or linker components with zero net voltage over every cycle. The contraction operation that substitutes vertices and edges, respectively, for dangling components and linkers yields a 3-connected graph with the same periodicity. 1-Periodic graphs can be analysed in the same way through their LQGs but the result may not be 3-connected. It is claimed that long-range topological properties of periodic graphs are respected by contraction so that contracted graphs can represent topological classes of crystal structures, be they rods, layers or three-dimensional frameworks.