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A general lower bound for the domination number of cylindrical graphs

José Juan Carreño, José Antonio Martínez, María Luz Puertas

Abstract

In this paper we present a lower bound for the domination number of the Cartesian product of a path and a cycle, that is tight if the length of the cycle is a multiple of five. This bound improves the natural lower bound obtained by using the domination number of the Cartesian product of two paths, that is the best one known so far.

A general lower bound for the domination number of cylindrical graphs

Abstract

In this paper we present a lower bound for the domination number of the Cartesian product of a path and a cycle, that is tight if the length of the cycle is a multiple of five. This bound improves the natural lower bound obtained by using the domination number of the Cartesian product of two paths, that is the best one known so far.

Paper Structure

This paper contains 5 sections, 5 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Known bounds and exact values
  3. Wasted domination
  4. The lower bound
  5. Conclusions

Key Result

Theorem 1

Let $S_{ij}^k$ be the set of all paths of length $k$ from $v_i$ to $v_j$ in $G$ and let $A(G)$ be the matrix defined by If $A^k$ is the $k$-th $(\min, +)$ power of $A$, then $(A(G)^k)_{ij}=\min \{\ell(Q)\colon Q\in S_{ij}^k\}$.

Figures (3)

  • Figure 1: The cylinder $P_m \Box C_n$
  • Figure 2: Dominating set of $P_{12} \Box C_{10}$ with $\frac{(12+2)10}{5}$ vertices and following a regular pattern
  • Figure 3: Partition of the cylinder $P_m \Box C_n$

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Corollary 1