Decomposing infinite graphs

In his classic paper Über eine Eigenschaft der ebenen Komplexe, Wagner [ 19 ] tackles the following problem. Kuratowski’s theorem, in its excluded minor version, states that a finite graph is planar if and only if it has no minor isomorphic to K5 or to K3,3. (A minor of G is any graph obtained from some H ⊂ G by contracting connected subgraphs.) If we exclude only one of these two minors, the graph may no longer be planar—but will it be very different from a planar graph? For example, can the non-planarity of an arbitrary finite graph without a K5 minor be tied down to certain parts of it, the rest of the graph being planar? Wagner’s solution to this problem is based on the following observation. Suppose we take two graphs G1 and G2, neither of which has a K5 minor, and paste them together along a complete subgraph. (Following Wagner, we shall use the term simplex for complete graphs. So here we let G = G1 ∪G2 and assume that G1 ∩G2 is a simplex.) Then the resulting graph G is again K5-free (has no K5 minor). For if H1, . . . , H5 are connected subgraphs of some H ⊂ G whose contraction yields a K5, then either G1 or G2 must also have such subgraphs (Fig. 1), contrary to our assumption that these graphs are K5-free.