The Minimum Dominating Set (MDS) problem is one of the well-studied problems in computer science. It is well-known that this problem is \(\mathsf {NP}\)-hard for simple geometric objects; unit disks, axis-parallel unit squares, and axis-parallel rectangles to name a few. An interesting variation of the MDS problem with rectangles is when there exists a straight line that intersects each of the given rectangles. In the recent past researchers have studied the maximum independent set, minimum hitting set problems on this setting with different geometric objects. We study the MDS problem with axis-parallel rectangles, unit-height rectangles, and unit squares in the plane. These geometric objects are constrained to be intersected by a straight line. For axis-parallel rectangles, we prove that this problem is \(\mathsf {NP}\)-hard. When the objects are axis-parallel unit squares, we present a polynomial time algorithm using dynamic programming. We provide a polynomial time algorithm for unit-height rectangles as well. For unit squares that touch the straight line at a single point from either side of the straight line, we show that there is an \(O(n\log n)\)-time algorithm.

最小支配集（MDS）问题是计算机科学中经过充分研究的问题之一。众所周知，对于简单的几何对象，此问题是\（\ mathsf {NP} \）-很难解决。单位磁盘，平行轴单位正方形和平行轴矩形等。带有矩形的MDS问题的一个有趣变化是，当存在一条与每个给定矩形相交的直线时。最近，研究人员研究了在不同几何对象的情况下，最大独立集，最小命中集的问题。我们研究MDS平面中轴平行矩形，单位高度矩形和单位正方形的问题。这些几何对象被约束为与直线相交。对于平行于轴的矩形，我们证明此问题是\（\ mathsf {NP} \）- hard。当对象是轴平行的单位平方时，我们提出使用动态规划的多项式时间算法。我们还为单位高度矩形提供了多项式时间算法。对于从直线的任一侧在单个点处接触直线的单位平方，我们表明存在\（O（n \ log n）\） -time算法。