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The 2-domination number of cylindrical graphs

José Antonio Martínez, Ana Belén Castaño-Fernández, María Luz Puertas

TL;DR

We study the $2$-domination number $\gamma_2$ of cylinders $P_m Box C_n$, focusing on large $m$ and $n$ with $n$ a multiple of 3. The authors derive a border-$2$-dominating lower bound using wasted domination and a tropical (min,+) matrix-product framework to control the auxiliary quantity $\omega_2(n)$, and they provide a constructive upper bound pattern yielding exact values in key regimes. A central contribution is a computational pipeline based on a digraph $D$ whose arcs carry a label $\ell$; this enables $(min,+)$-powers of $A(D)$ to determine $\omega_2(n)$ and thus $\gamma_2(P_m Box C_n)$ for large $n$, with explicit results such as $\omega_2(n)=2n$ for $n\ge 45$ and special cases $n=16,19$. The main finding is that for $m\ge 8$ and $n$ a multiple of 3, $\gamma_2(P_m Box C_n)=\frac{(m+2)n}{3}$, demonstrating how border-based domination techniques and tropical algebra jointly solve $2$-domination on cylinders and highlighting computational aspects for broader parameter ranges.

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.

The 2-domination number of cylindrical graphs

TL;DR

We study the -domination number of cylinders , focusing on large and with a multiple of 3. The authors derive a border--dominating lower bound using wasted domination and a tropical (min,+) matrix-product framework to control the auxiliary quantity , and they provide a constructive upper bound pattern yielding exact values in key regimes. A central contribution is a computational pipeline based on a digraph whose arcs carry a label ; this enables -powers of to determine and thus for large , with explicit results such as for and special cases . The main finding is that for and a multiple of 3, , demonstrating how border-based domination techniques and tropical algebra jointly solve -domination on cylinders and highlighting computational aspects for broader parameter ranges.

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 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.
Paper Structure (8 sections, 13 theorems, 33 equations, 4 figures, 1 algorithm)

This paper contains 8 sections, 13 theorems, 33 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. A lower bound of the 2-domination number of cylinders
  3. Proof of Lemma \ref{['lemma:third']}
  4. Construction of digraph $\mathcal{D}$
  5. Definition of the arc labeling $\ell$
  6. The $(\min ,+)$ powers of the matrix $A(\mathcal{D})$
  7. Proof of the lemma
  8. Conclusions

Key Result

Lemma 1

Let $S$ be a $2$-dominating set of $P_m\Box C_n$, then

Figures (4)

  • Figure 1: The set of black vertices $2$-dominates the infinite grid
  • Figure 2: Rows and columns of $P_m\Box C_n$
  • Figure 3: Black and gray vertices $2$-dominates $P_{13}\Box C_{18}$
  • Figure 4: Newly dominated vertices

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Definition 2
  • Lemma 3
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 21 more