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. The stab number of a graph G, denoted by stab(G), is the minimum integer k such that G is a k-SRIG. In this paper, we introduce some natural subclasses of 2-SRIG and study the containment relationships among them. We also give a linear time recognition algorithm for one of those classes. In this paper, we prove that the Chromatic Number problem is NP-complete even for 2-SRIGs. This strengthens a result by Imai and Asano . We also show that triangle-free 2-SRIGs are three colorable.
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