General · Networks · manipulative

Planar drawings and V − E + F = 2

Start with a clean planar drawing, then tangle or untangle by dragging. Toggle V, E, and F counts onto the diagram — including the outer face.

Drag vertices — or use “crossed start” to practise untangling

Red × marks show crossings · dashed border hints at the infinite outer face

Graph

Each preset is a connected planar graph — switch to see V − E + F = 2 with different counts.

Drawing

Show on diagram

Vertices V

count badge + vertex labels

Edges E

number each edge mid-span

Faces F

face numbers when the drawing is planar

Counts

Working