MathLabs
Step 1 of 6: Puncture one face and flatten the polyhedron onto the plane
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In plain words

Cut away one face and regard the remaining surface as a topological disk. We can redraw its vertices and edges in the plane without crossings, preserving incidences; this is a planar embedding, not a claim that every physical net can be laid flat without overlap.

Vgraph=V,Egraph=E,Fgraph=(F−1)+1=FV_{\text{graph}} = V, \quad E_{\text{graph}} = E, \quad F_{\text{graph}} = (F-1) + 1 = F
A cube with one face marked before it is removed
A cube drawn in 3D with one square face shaded to mark it as the face that will be removed so the rest of the surface can be flattened into a planar graph.
Detailed analysis

Pick any face of the convex polyhedron, remove it, and stretch the remaining surface flat onto the plane like an open net. This works because the surface of a convex polyhedron is topologically a sphere, so removing one face leaves a disk that can be flattened without any edges crossing. The result is a connected planar graph with exactly the same vertices VV and edges EE as the polyhedron; the F−1F-1 remaining faces become the F−1F-1 bounded regions of the graph, and the removed face becomes the single unbounded outer region, so the graph still has FF faces in total.

Terms in this step
Planar graph
A graph (vertices joined by edges) that can be drawn in the plane so that no two edges cross.
Topological sphere
A surface that can be continuously stretched and deformed (without cutting or gluing) into an ordinary round sphere; a cube's surface qualifies, a doughnut's does not.
Knowledge used in this step
Polyhedra and their volumesPlanar graphs
Common mistake. This flattening step needs the polyhedron's surface to be a topological sphere. For a polyhedron with a hole through it — Lakatos's classic 'picture frame' counterexample — removing a face leaves a surface that cannot be flattened without edges crossing, and indeed V−E+F=0V - E + F = 0 for that shape, not 22.