Summary: We introduce the notion of stab number and exact stab number of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. 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 chosen such that each rectangle is intersected by at least one of them. If there exists such a representation with the additional property that each rectangle intersects exactly one of the \(k\) horizontal lines, then the graph \(G\) is said to be a \(k\)-exactly stabbable rectangle intersection graph, or \(k\)-ESRIG for short. The stab number of a graph \(G\), denoted by \(\mathrm{stab}(G)\), is the minimum integer \(k\) such that \(G\) is a \(k\)-SRIG. Similarly, the exact stab number of a graph \(G\), denoted by \(\mathrm{estab}(G)\), is the minimum integer \(k\) such that \(G\) is a \(k\)-ESRIG.
In this work, we study the stab number and exact stab number of some subclasses of rectangle intersection graphs. A lower bound on the stab number of rectangle intersection graphs in terms of its pathwidth and clique number is shown. Tight upper bounds on the exact stab number of split graphs with boxicity at most 2 and block graphs are also given. We show that for \(k\leq 3,k\)-SRIG is equivalent to \(k\)-ESRIG and for any \(k\geq 10\), there is a tree which is a \(k\)-SRIG but not a \(k\)-ESRIG.
We also develop a forbidden structure characterization for block graphs that are 2-ESRIG and trees that are 3-ESRIG, which lead to polynomial-time recognition algorithms for these two classes of graphs. These forbidden structures are natural generalizations of asteroidal triples. Finally, we construct examples to show that these forbidden structures are not sufficient to characterize block graphs that are 3-SRIG or trees that are \(k\)-SRIG for any \(k\geq 4\).