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arXiv · 2409.16703

The 2-domination number of cylindrical graphs

José Antonio Martínez·Ana Belén Castaño-Fernández·María Luz Puertas
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Abstract

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.

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José Antonio Martínez, Ana Belén Castaño-Fernández, María Luz Puertas. 2024-09-25. The 2-domination number of cylindrical graphs. https://doi.org/10.1007/s40314-022-02137-1

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