We consider arrangements of axis-aligned rectangles in the plane. A geometricarrangement specifies the coordinates of all rectangles, while a combinatorialarrangement specifies only the respective intersection type in which each pairof rectangles intersects. First, we investigate combinatorial contactarrangements, i.e., arrangements of interior-disjoint rectangles, with atriangle-free intersection graph. We show that such rectangle arrangements arein bijection with the 4-orientations of an underlying planar multigraph andprove that there is a corresponding geometric rectangle contact arrangement.Moreover, we prove that every triangle-free planar graph is the contact graphof such an arrangement. Secondly, we introduce the question whether a givenrectangle arrangement has a combinatorially equivalent square arrangement. Inaddition to some necessary conditions and counterexamples, we show thatrectangle arrangements pierced by a horizontal line are squarable under certainsufficient conditions.
展开▼
