New worlds for Pac-Man - Liceo Scientifico R. Bruni (Padova)

Inside a regular polygon with 2k sides we must place n cherries and paths (not necessarily segments) that connect them. Each pair of paths can only cross in cherries and there can be no self-crossing.
For k = 2, 3, and 4, we want to figure out the maximum number of paths we can fit into our version of PacMan in each of the following cases.
(a) whether the sides of the polygon are barriers to the passage of PacMan;
(b) if PacMan's exit from one side [AB] of the polygon corresponds to entry into the opposite side [CD] ([AD] and [BC] being diagonals of maximum length), such that the distance from A of the exit point is the same distance from C to the entry point.