ORIGINAL THREE-BY-THREE NET TEACHING EXAMPLE
Tiles: A B C / D E F / G H I. Center E is orange. Nonwrapping boundaries.
This is an original teaching network, not Simon Tatham's linked 5x5 game.
Answer connections: A-B, B-C, B-E, E-D, E-F, D-G, F-I, I-H.
Nine vertices; eight edges; every vertex reachable from E; no loops.
Tile shapes: A endpoint; B T; C endpoint; D elbow; E T; F elbow; G endpoint; H endpoint; I elbow.
Reasoning: B is a T in top-middle so must face left, right and down.
A and C must connect to B. G can point up or right but H is another endpoint so G-H would form a disconnected pair. G must point up to D. D's elbow must connect down to G and right to E. E must accept from B and D. I's bottom-right elbow must point up to F and left to H. F needs down to I and its perpendicular elbow branch cannot leave the right edge, so it points left to E. This forces E's third branch right to F. H points right to I.