A vertex subset S of a graph G is said to 2-dominate the graph if each vertex not in S has at least two neighbors in it. As usual, the associated parameter is the minimum cardinal of a 2-dominating set, which is called the 2-domination number of the graph G. We present both lower and upper bounds of the 2-domination number of cylinders, which are the Cartesian products of a path and a cycle. These bounds allow us to compute the exact value of the 2-domination number of cylinders where the path is arbitrary, and the order of the cycle is n $\equiv$ 0(mod 3) and as large as desired. In the case of the lower bound, we adapt the technique of the wasted domination to this parameter and we use the so-called tropical matrix product to obtain the desired bound. Moreover, we provide a regular patterned construction of a minimum 2-dominating set in the cylinders having the mentioned cycle order.

翻译：若图G的顶点子集S满足：每个不在S中的顶点在S中至少有两个邻点，则称S 2-控制该图。通常，对应的参数是2-控制集的最小基数，称为图G的2-控制数。本文给出了柱面（即一条路与一个圈的笛卡尔积）的2-控制数的上下界。这些界使得我们能够计算柱面2-控制数的精确值，其中路是任意的，而圈的长度n满足n $\equiv$ 0(mod 3)且可任意大。在下界情形中，我们将“浪费控制”技术适配于此参数，并利用所谓的热带矩阵积来获得所需的下界。此外，对于具有上述圈长的柱面，我们给出了最小2-控制集的一种规则模式构造。