Let $P$ be a given polygon in $\mathbb{R}^{2}$ such that all the vertices lie on integral points $\mathbb{Z}^{2}$. An integral polygonal sub-divison of $P$ is a subdivison of $P$ into integral sub-polygons. I would like to know:

If there is an algorithm to generate all such sub-divisons? I would prefer an easier to implement (possibly less efficient) algorithm over a more efficient one. I do not want probabilistic algorithms unless they have a very high efficiency.
To illustrate the problem here are some pictures:

The initial polygon is like this:

Integral polygon

An allowed sub-divison

A valid Integral Subdivison

An invalid sub-divison

Not a Valid Integral Subdivison

Thank you in advance.

  • 1
    $\begingroup$ Is P convex? What is an integral sub-polygon? What is a polygonal division? Perhaps you should clarify... $\endgroup$
    – Aryabhata
    May 7, 2015 at 23:09
  • $\begingroup$ 1. Please edit the question with clarifications, as suggested by Aryabhata. 2. In addition, what have you tried? There seems to be a straightforward recursive algorithm: it might inefficient, but it will be simple and meet your requirements. We expect you to make a significant effort on your own before asking, and to show us in the question what you've tried. $\endgroup$
    – D.W.
    Jun 5, 2015 at 0:51


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