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Proving exact values for the $2$-limited broadcast domination number on grid graphs

Aaron Slobodin, Gary MacGillivray, Wendy Myrvold

TL;DR

The method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Abstract

We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Proving exact values for the $2$-limited broadcast domination number on grid graphs

TL;DR

The method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Abstract

We establish exact values for the -limited broadcast domination number of various grid graphs, in particular for and all , for all , and for and all . We also produce periodically optimal values for and for , for , and for . Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Paper Structure

This paper contains 12 sections, 15 theorems, 31 equations, 5 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Intuition and Definitions
  3. Eliminating Possible Induced Sub-broadcasts of Fixed Cost
  4. Domination Requirement
  5. Forbidden Broadcasts
  6. Optimality Requirement
  7. Proof by Induction
  8. Necessary Broadcasts
  9. Considering All Possible Sub-Cases
  10. Algorithm to Prove Lower Bounds
  11. Results
  12. Future Work

Key Result

Proposition 2.3

slobodinfinal For $n \geq 3$,

Figures (5)

  • Figure 1: The graph $H_{m,k}$ with columns labelled $c_1, c_2, \dots, c_k$, $C$ indicated by the thick black rectangle, and possible broadcast vertices exterior to $C$ which can only dominate vertices in columns $c_5, c_6, c_{k-5},$ and $c_{k-4}$.
  • Figure 2: Forbidden broadcasts.
  • Figure 3: Assumed sub-broadcast $g$ induced by $V(C)$ of cost 7.
  • Figure 4: (Left & Middle) Procedure which reduces $G_{m,n_0}$ to $G_{m,n_0-4}$. (Right) Broadcast of cost 3 which dominates $R'$.
  • Figure 5: The graph $H_{m,k}$ with columns labelled $c_1, c_2, \dots, c_k$ and $C$ indicated by the thick black rectangle. (Left) Resulting necessary broadcasts of strength 2 in columns $c_4$ and $c_{k-3}$ forced by vertices undominated in columns $c_6$ and $c_{k-5}$. (Right) Necessary broadcast $g'$ as described in Section \ref{['subsec:necbroad']}, the vertex undominated by $g'$ indicated by the green circle, and one of the five possible sub-broadcasts $g"$ which extend $g'$ to dominate the vertex undominated by $g'$ in column $c_{k-4}$.

Theorems & Definitions (36)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 3.1
  • proof
  • proof
  • Example 3.2
  • ...and 26 more