We study two decomposition problems in combinatorial geometry. The first part
deals with the decomposition of multiple coverings of the plane. We say that a
planar set is cover-decomposable if there is a constant m such that any m-fold
covering of the plane with its translates is decomposable into two disjoint
coverings of the whole plane. Pach conjectured that every convex set is
cover-decomposable. We verify his conjecture for polygons. Moreover, if m is
large enough, we prove that any m-fold covering can even be decomposed into k
coverings. Then we show that the situation is exactly the opposite in 3
dimensions, for any polyhedron and any m we construct an m-fold covering of
the space that is not decomposable. We also give constructions that show that
concave polygons are usually not cover-decomposable. We start the first part
with a detailed survey of all results on the cover-decomposability of polygons.
The second part investigates another geometric partition problem, related to
planar representation of graphs. The slope number of a graph G is the smallest
number s with the property that G has a straight-line drawing with edges of at
most s distinct slopes and with no bends. We examine the slope number of
bounded degree graphs. Our main results are that if the maximum degree is at
least 5, then the slope number tends to infinity as the number of vertices
grows but every graph with maximum degree at most 3 can be embedded with only
five slopes. We also prove that such an embedding exists for the related notion
called slope parameter. Finally, we study the planar slope number, defined only
for planar graphs as the smallest number s with the property that the graph has
a straight-line drawing in the plane without any crossings such that the edges
are segments of only s distinct slopes. We show that the planar slope number of
planar graphs with bounded degree is bounded.Comment: This is my PhD thesi
