Fisunenko A. Constructing a generator of geometric objects with defined properties on the plain.

The thesis considers several interconnected problems that relate to constructing, counting and randomly generating sets of different types of simple polygons, which embrace all given input points as their vertices and satisfy defined criteria. For analysis of input point sets and constructing simple polygons, we introduce a star partitioning equivalence diagram and points mutual visibility graph and devise their properties. For solving problems of constructing simple polygons, counting a set of all possible simple polygons and generating random polygons on this set, we use a method of incremental construction of polygonal chain with backtracking. The method assumes cutting unproductive branches of variants' tree by analyzing a structure of points' mutual visibility graph, which is a geometrical graph in fact. We extend the list of necessary and sufficient conditions for existence of Hamiltonian path in such graphs both by analyzing graph's connectedness and by using specific geometric conditions for constructing simple polygons. For analysis of hamiltonicity of the graph, we also use its biconnected components and articulation points. We formulate and prove several propositions allowing elimination of redundant edges from the graph, what leads to decreasing number of traverses in variants' tree.