Polygonal paths and polygons with prescribed set of vertices Let $A$ be a finite set of points in the plane. How can we determine if there is a simple open polygonal path (i.e. without intersections),  whose vertices are exactly $A$, with no straight angles between adjacent sides?   Since in  mathoverflow.net/q/226469/4312 question is  only about cycles,  the stronger question also remains. Namely - how  to determine if there exists a polygon with non-intersecting sides and  without straight angles, whose vertices are exacly $A$ ?
Particular interesting case is when $A$ is a set of points $(x;y)$ with $0\leq x+y \leq 2n$ , where $x,y$ are nonnegative integers. Hypothesis : no for path for $n=1$ and $n=2$.
 A: The following polygon is a counterexample to the hypothesis for $n=4$. Namely we consider the set of integer  points $(x,y)|0\le x+y\le 8$. The picture is on a square-lined paper, where the size of one square is $\frac{1}{2}\times \frac{1}{2}$
The picture was constructed in collaboration with Svetlana Ermakova.
So my guess that it will be possible to do the same thing for all $n\ge 4$

A: Not an answer, just an illustration for $6$ points, $0 \le x+y \le 2$.

     




*

*$A,B,C$: Point $3$ cannot connect to $1$ or $6$, so it must connect to
$2,5$ or $4,5$ or $2,4$.

*$B$: Point $6$ is now isolated by $34$ from $1$ and $2$.

*$C$: Point $1$ is now isolated by $34$ from $5$ and $6$.

*$D$: $1$ is trapped.

*$E$: $2$ is trapped.



(Later.)
Here is a simple polygonal path through the $28$ lattice points
$0 \le x+y \le 6$:

     

A: This paper addresses similar (but I don't think identical) questions.
In any case, a key search phrase is covering path.

Dumitrescu, Adrian, Dániel Gerbner, Balázs Keszegh, and Csaba D. Tóth. "Covering paths for planar point sets." Discrete & Computational Geometry 51, no. 2 (2014): 462-484. Journal link.



What appears to be unique is the OP's insistence that there is a turn
at every vertex--no collinearities.
Of course, if the points are in general position, there is
automatically a turn at every vertex.

          


          

Fig.2


