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Sherry 1.3 GMA3150 32bit Fix2.7zQ:

Checking if a polygon contains all points of a given order

Let $D=\{d_0, d_1, \cdots,d_m\}$ be a set of $m+1$ points in a plane that are ordered counterclockwise (example: $d_0=(0,0)$, $d_1=(1,0)$, $d_2=(0,1)$, and so on). Let $S_m$ be a convex polygon with vertices $a_i=d_{i-1}+d_i$ ($i=1,2,\cdots,m$).
Consider the following problem: Given a set of $m$ points in a plane, check if $S_m$ contains the set $D$.
My idea is, first, to check if $S_m$ contains only one of the vertices of $D$. If not, then $S_m$ can’t contain $D$. If $S_m$ contains one of the vertices, then, there might be another vertex inside $S_m$ that is not within $S_m$’s boundary, and we can proceed by moving from one vertex to its neighbor. We can do this until all points of $D$ are checked to see if they are inside $S_m$.
If the problem is in the class of P problems, then this approach is correct. But, if not, does somebody know how this problem can be solved?

A:

If there are several disjoint convex polygons then the problem is NP-complete.
The proof is straightforward.

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