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 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 ≤ 3, k-SRIG is equivalent to k-ESRIG and for any k ≥ 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 ≥ 4.

中文翻译：

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关于矩形相交图的刺数

我们介绍的概念刺数目和精确刺数目矩形相交曲线图中，否则被称为boxicity的曲线图至多2，一种图形ģ被说成是一个ķ - stabbable矩形相交曲线图，或ķ - SRIG为短，如果它具有矩形交点表示，其中可以选择k条水平线，以使每个矩形与其中至少一条相交。如果存在具有附加属性的表示形式，即每个矩形恰好与k条水平线之一相交，则将图形G称为k-恰好stabbable矩形相交曲线图，或ķ - ESRIG的简称。由s t a b（G）表示的图G的刺数是最小整数k，使得G为k -SRIG。类似地，由e s t a b（G）表示的图G的确切刺破数是最小整数k，使得G为k-ESRIG。在这项工作中，我们研究矩形相交图的某些子类的刺数和精确刺数。显示了矩形相交图的刺数的下限，即其路径宽度和集团数。还给出了具有最多2个方块性的分裂图的精确刺数的严格上限。我们表明，ķ ≤3，ķ -SRIG相当于ķ -ESRIG和任何ķ ≥10，有一棵树是一个ķ -SRIG但不是ķ-ESRIG。我们还为2-ESRIG的框图和3-ESRIG的树开发了禁止结构表征，这导致了这两类图的多项式时间识别算法。这些禁止的结构是小行星三元组的自然概括。最后，我们构建的例子来显示，这些禁止结构不足以表征块的图表，是3- SRIG或者是树木ķ任何-SRIG ķ ≥4。