On the $2$-domination number of cylinders with small cycles

TL;DR

This paper computes the 2-domination number for cylinders formed by small cycles and paths, using algorithms based on the $( ext{min},+)$ matrix product, and proposes a conjecture for general formulas.

Contribution

It introduces algorithms to determine the 2-domination number in cylinders with small cycles and conjectures a general formula for this parameter.

Findings

Computed $oldsymbol{eta_2(C_n imes P_m)}$ for $3 \\leq n \\leq 15$, $m \\geq 2$.

Developed algorithms involving the $( ext{min},+)$ matrix product.

Proposed a conjecture for the general formula of the 2-domination number in cylinders.

Abstract

Domination-type parameters are difficult to manage in Cartesian product graphs and there is usually no general relationship between the parameter in both factors and in the product graph. This is the situation of the domination number, the Roman domination number or the $2$-domination number, among others. Contrary to what happens with the domination number and the Roman domination number, the $2$-domination number remains unknown in cylinders, that is, the Cartesian product of a cycle and a path and in this paper, we will compute this parameter in the cylinders with small cycles. We will develop two algorithms involving the $(\min,+)$ matrix product that will allow us to compute the desired values of $\gamma_2(C_n\Box P_m)$, with $3\leq n\leq 15$ and $m\geq 2$. We will also pose a conjecture about the general formulae for the $2$-domination number in this graph class.