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Private neighbors, perfect codes and their relation with the $\vt$-number of closed neighborhood ideals

Delio Jaramillo-Velez, Hiram H. López, Rodrigo San-José

Abstract

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\vt$-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the $\vt$-number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the $\vt$-number is a lower bound for the regularity of bipartite and well-covered graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the $\vt$-number of some special family of graphs.

Private neighbors, perfect codes and their relation with the $\vt$-number of closed neighborhood ideals

Abstract

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the -number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the -number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the -number is a lower bound for the regularity of bipartite and well-covered graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the -number of some special family of graphs.

Paper Structure

This paper contains 11 sections, 14 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theory
  4. Private neighbors
  5. Dominating and irredundant sets
  6. Commutative algebra
  7. Coding theory
  8. The $\mathtt{v}$-number of neighborhood ideals
  9. The relation between the $\mathtt{v}$-number and other invariants
  10. The $\mathtt{v}$-number and perfect codes
  11. Code for examples

Key Result

Proposition 2.2

Every maximal independent set is a minimal dominating set.

Figures (2)

  • Figure 1: The vertices in red compose the Hamming code $\mathcal{H}_2(2)$.
  • Figure 2: Two close neighborhoods of elements in $\mathcal{H}_2(3)$.

Theorems & Definitions (37)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: huffman_codes
  • Proposition 3.1: irred-decom-NG
  • Proposition 3.2: intro_private_nei
  • proof
  • Remark 3.3
  • Remark 3.4
  • ...and 27 more