  We introduce the notion of \emph{stab number} and \emph{exact stab number} of
rectangle intersection graphs, otherwise known as graphs of boxicity at most 2.
A graph $G$ is said to be a \emph{$k$-stabbable rectangle intersection graph},
or \emph{$k$-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 \emph{$k$-exactly
stabbable rectangle intersection graph}, or \emph{$k$-ESRIG} for short. The
stab number of a graph $G$, denoted by $stab(G)$, is the minimum integer $k$
such that $G$ is a $k$-SRIG. Similarly, the exact stab number of a graph $G$,
denoted by $estab(G)$, 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 $k\geq 4$.
