[Paper Review] New Upper Bounds on the Distance Domination Numbers of Grids

Armando Grez, Michael Farina|arXiv (Cornell University)|Oct 15, 2014

Advanced Graph Theory Research8 references3 citations

TL;DR

This paper improves upper bounds for the k-distance domination number of m×n grid graphs by refining a lattice-based construction using a ring homomorphism φₖ: ℤ² → ℤₚ, where p = 2k² + 2k + 1. It proves that for sufficiently large m and n (both > 2p), the k-distance domination number is at most ⌊(m+2k)(n+2k)/p⌋ − 4, significantly reducing prior bounds from Fata, Smith, and Sundaram (2013).

ABSTRACT

In his 1992 Ph.D. thesis Chang identified an efficient way to dominate $m imes n$ grid graphs and conjectured that his construction gives the most efficient dominating sets for relatively large grids. In 2011 Gonçalves, Pinlou, Rao, and Thomassé proved Chang's conjecture, establishing a closed formula for the domination number of a grid. In March 2013 Fata, Smith and Sundaram established upper bounds for the $k$-distance domination numbers of grid graphs by generalizing Chang's construction of dominating sets to $k$-distance dominating sets. In this paper we improve the upper bounds established by Fata, Smith, and Sundaram for the $k$-distance domination numbers of grids.

Motivation & Objective

• Address the lack of tight upper bounds for k-distance domination numbers in grid graphs beyond k=2.
• Improve upon the 2013 upper bound for γₖ(Gₘ,ₙ) established by Fata, Smith, and Sundaram.
• Provide a general construction applicable to all k ≥ 1 using algebraic structure in ℤ².
• Establish a tighter asymptotic upper bound that reduces the size of dominating sets by removing four vertices per corner.
• Confirm that the improved bound holds when m and n exceed 2(2k² + 2k + 1), ensuring non-overlapping corner removals.

Proposed method

• Embed the m×n grid Gₘ,ₙ into the integer lattice ℤ² and define its k-distance neighborhood Yₘ₊₂ₖ,ₙ₊₂ₖ.
• Use a ring homomorphism φₖ: ℤ² → ℤₚ, where p = 2k² + 2k + 1, to define periodic dominating sets as inverse images φₖ⁻¹(ℓ̄).
• Prove via combinatorial geometry that for some ℓ̄ ∈ ℤₚ, the set φₖ⁻¹(ℓ̄) ∩ Yₘ₊₂ₖ,ₙ₊₂ₖ has size at most ⌊(m+2k)(n+2k)/p⌋.
• Apply geometric transformations (shifts) to vertices in the dominating set to remove at least one vertex from each of the four corners of the set.
• Use slope-based case analysis (based on lines L₁ and L₂) to justify corner vertex removals without losing domination of the grid.
• Move all vertices outside Gₘ,ₙ to their nearest neighbors inside Gₘ,ₙ to ensure the final dominating set is contained within the grid.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound for the k-distance domination number of an m×n grid graph for k ≥ 3?
• RQ2 How can algebraic structures in ℤ² be leveraged to construct efficient k-distance dominating sets?
• RQ3 Can the size of the dominating set be reduced by removing vertices from the corners of the lattice-based construction?
• RQ4 Under what conditions on m and n can corner vertices be safely removed without compromising domination?
• RQ5 Does the improved bound ⌊(m+2k)(n+2k)/p⌋ − 4 hold for all sufficiently large m and n, and how does it compare to prior bounds?

Key findings

• The paper establishes a new upper bound for the k-distance domination number of m×n grid graphs: γₖ(Gₘ,ₙ) ≤ ⌊(m+2k)(n+2k)/p⌋ − 4, where p = 2k² + 2k + 1.
• For k=3, the new bound reduces the upper estimate from 139 to 128 for G₅₁,₅₂, a 9.4% improvement over the prior bound.
• The improvement is consistent across large grids: for G₆₅,₆₆, the new bound is 200 compared to the old bound of 211, a 5.2% reduction.
• Corner vertex removal is proven feasible via geometric shifts in three distinct slope-based cases, ensuring domination is preserved.
• The bound is valid when m and n are both greater than 2p = 2(2k² + 2k + 1), ensuring non-overlapping corner adjustments.
• By removing at least four vertices from the lattice-based dominating set (one per corner), the construction achieves a tighter bound than prior methods.