Not an answer, just an illustration for $6$ points, $0 \le x+y \le 2$. <hr /> [![Poly6][1]][1] <hr /> - $A,B$: 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. [1]: https://i.sstatic.net/ZJuR1.jpg