Strings and coins
Strings and coins is a form of the game dots and boxes represented by its dual graph. Two players take turns cutting strings. Whenever a player frees one or more coins, they keep them and cut again, aiming to collect more coins than their opponent. The S&C coins correspond to the D&B boxes, and the S&C edges correspond to the D&B edges (but roughly perpendicular). In this version the graph is somewhat random rather than a square grid, which would correspond to having triangles, pentagons, etc for boxes. Also it works like the "Swedish" version of dots and boxes, ie where the perimeter edges are already added. Strings and coins would also allow non-plane graphs, ie with crossing edges, which I don't think have an equivalent in dots and boxes.