# Combinatorial game similar to Sprouts

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?

Basically this game is "Sprouts without midpoints". One starts with $n$ points in the plane. Then a move consists of joining two points (it is also allowed to join a point with itself, i.e. to make loops). The lines are not allowed to intersect, i.e. the graph should be planar (otherwise the game would be too easy to solve). The degree of each point is supposed to be $\leq 3$. Thus, points of degree $3$ are "dead". The player with the last move wins.

I have already determined some game outcomes, but wanted to know a reference. On the german Wikipedia it says that the sprouts variant where the players may decide if they add a midpoint is already solved (the first player wins) and "known" as black-and-white sprouts, but I could not find anything about this, and also this game has different game outcomes than the game described above.

• PS: It should be clear that this game is much simpler than classical Sprouts. – HeinrichD Dec 28 '16 at 9:46
• Parallel edges are allowed as well? – Klaus Draeger Dec 28 '16 at 11:31
• @KlausDraeger: Yes, they are allowed. – HeinrichD Dec 28 '16 at 11:36
• Can you join a point to itself as in usual sprouts? I assume so. – znt Dec 28 '16 at 13:31
• @znt: This is explicitly mentioned in the second paragraph. – HeinrichD Dec 28 '16 at 13:35