Not an answer, just an illustration for $6$ points, $0 \le x+y \le 2$.
[![Poly6][1]][1]
- $A,B$: Point $3$ cannot connect to $1$ or $6$, so it must connect to $2,5$ or $4,5$ ($2,4$ is symmetric).
- $B$: Point $6$ is now isolated by $34$.
- $C,D$: $5$ can connect to $4$ or $1$ (but not $6$).
- $C$: $1$ is trapped.
- $D$: $2$ is trapped.