Plexus Via Midpoints And Endpoints

This is a game for two players.

Start with a lattice of n-by-n dots (which are the vertices of an (n-1)-by-(n-1)-squares grid). n is odd.

One of the players draw a straight line-segment from the middle dot of the lattice to any adjacent dot above, below, left of, or right of the central dot.
(It doesn't matter which direction this line-segment is drawn.)

The game consists of "full-moves", each of which consists of two "half-moves". In a full-move, one player moves then the other. The order of the players in the full-moves alternates.
So, we have the players moving like this:
(1,2),(2,1),(1,2),(2,1),(1,2),(2,1),...

In a full-move, the players each draw a straight line-segment from a dot already connected with another dot via a line-segment, to an adjacent dot (in the direction of either above, below, left of, or right of). The adjacent dot must have no line-segments yet connected with it.

The first player to move within a full-move decides if his/her line segment will be drawn from a dot already connected to 2 or 3 line-segments (a midpoint), or from a dot already connected to by exactly one line-segment (an endpoint).

The second player to move in the full-move must also draw his/her line-segment from a midpoint or from an endpoint just as the first player in the full-move did.
In other words, if the first player in the full-move draws from a midpoint, the second player must also draw from a midpoint. And if the first player drew from an endpoint, the second player must also draw from an endpoint.
(A player is free to decide which midpoint or endpoint to draw from, however, provided that the dot he/she draws to doesn't have any line segments connecting to it yet.)

If it is impossible for the second player in the full-move to do as the first player did (draw from midpoint or draw from endpoint), he/she loses his turn in this particular full-move, and the first player to move in that full-move then gets a point. (The second player still moves first in the next full-move, though.)

The game continues until all dots have each been connected with at least one line-segment.

The player with the largest score wins.

Thanks,
Leroy Quet