On rectangle intersection graphs with stab number at most two

Rectangle intersection graphs are the intersection graphs of axis-parallel rectangles in the plane. A graph G is said to be a k-stabbable rectangle intersection graph, or k-SRIG for short, if it has a rectangle intersection representation in which k horizontal lines can be placed such that each rectangle intersects at least one of them. In this article, we introduce some natural subclasses of 2-SRIG and study the containment relationships among them. It is shown that one of these subclasses can be recognized in linear-time if the input graphs are restricted to be triangle-free. We also make observations about the chromatic number of 2-SRIGs. It is shown that the Chromatic Number problem is NP-complete for 2-SRIGs, by showing that the problem is NP-complete for 2-row B0-VPGs. This is a strengthening of some known results from the literature.