The 2-domination number of cylindrical graphs

TL;DR

This paper establishes bounds and exact values for the 2-domination number in cylindrical graphs, specifically Cartesian products of paths and cycles, using novel techniques like wasted domination and tropical matrix products.

Contribution

It provides the first comprehensive bounds and exact values for the 2-domination number of cylinders, introducing new methods for constructing minimum 2-dominating sets.

Findings

Exact 2-domination number for cylinders with cycle length divisible by 3.

Lower and upper bounds for the 2-domination number of cylinders.

A regular patterned construction for minimum 2-dominating sets.

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.