Two vertices are marked A and B. One player, Short, claims edges and is trying to build a claimed road from A to B. The other, Cut, deletes edges and is trying to leave A and B with no road between them at all. Pick a side; the computer takes the other one and plays perfectly. Nobody can force a draw — every game ends with one of you finished.
Two numbers describe every position. Short's cheapest road is the smallest number of fresh edges Short would still have to claim to finish a route — when it hits zero, Short has won. Cut's thinnest cut is the smallest number of deletable edges that would separate A from B — when it hits zero, Cut has won. The highlighted edges on the board are one such road and one such cut. The computer is not following either number; it searches the whole game to the end. The numbers are here so you can see what it is looking at.
Every board sits in exactly one of three classes: Short wins whoever starts, Cut wins whoever starts, or whoever moves first wins. There is never a draw. Boards 8, 9 and 10 are the same eight vertices — 12 edges, then 13, then 14 — and they land in all three classes in turn. One brace at a time is the whole difference. The slider is ordered by how hard the board is to play, not by how big it looks: board 7 is the widest of the ten and three of them are harder.