Euler bounds prove both classical obstruction graphs are nonplanar. This is a compact note, but the quantifiers and hypotheses stay on the page.
Start locally
A planar embedding divides the sphere into vertices, edges, and faces. Euler's relation \(|V|-|E|+|F|=2\) constrains density, while the dual \(G^\ast\) records adjacency of faces.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
Compute before generalising
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
The global view
The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.
Edge conditions
Planarity is a property of a graph, while a plane graph includes a chosen embedding. The dual depends on that embedding.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.