Kuratowski's theorem

Any network can be laid out in 3D space. (This is related to the Whitney embedding theorem that any d-dimensional manifold can be embedded in (2d + 1)-dimensional space.) When one says that a network is planar what one means is that it can be laid out in ordinary 2D space without any lines crossing. Kuratowski's theorem that planarity is associated with the absence of specific subgraphs in a network is an important result in graph theory established in the late 1920s. A subgraph is formally defined to be what one gets by selecting just some subset of connections in a network—and with this definition Kuratowski's theorem must allow extensions of K_{5} and K_{3,3} where extra nodes have been inserted in the middle of connections. (K_{5} and K_{3,3} are examples of so-called complete graphs, obtained by taking sets of specified numbers of nodes and connecting them in all possible ways.) Another approach is to consider reducing whole networks to so-called minors by deleting connections or merging connected nodes, and in this case Wagner's theorem shows that any non-planar network must be exactly reducible to either K_{5} or K_{3,3}.

One can generalize the question of planarity to asking whether networks can be laid out on 2D surfaces with various topological structures—and in fact the genus of a graph can be defined to be the number of handles that must be added to a plane to embed the graph without crossings. But even on a torus it turns out that there is no finite set of (extended) subgraphs whose absence guarantees that a network can successfully be laid out. Nevertheless, if one considers minors a finite list does suffice—though for example on a torus it is known that at least 800 (and perhaps vastly more) are needed. (There is in fact a general theorem established since the 1980s that absolutely any list of networks—say for example ones that cannot be laid on a given surface—must actually in effect always all be reducible to some finite list of minors.) Note that finding the genus for a particular trivalent network is in general NP-complete.