[Paper Review] The Bondage Number of Mesh Networks

Fu-Tao Hu, Yongchang Cao+1 more|arXiv (Cornell University)|Sep 19, 2011

Advanced Graph Theory Research25 references3 citations

TL;DR

This paper determines the exact bondage numbers for mesh networks $P_n \times P_m$ with $m = 2, 3, 4$ and $n \geq 2$, establishing $b(P_n \times P_2) = 1$, $b(P_n \times P_3) = 1$ for $n \geq 4$, and $b(P_n \times P_4) = 1$ for $n \notin \{5,6,9\}$, with $b(G_{5,4}) = b(G_{9,4}) = 3$ and $b(G_{6,4}) = 2$. The results are derived through structural analysis of dominating sets and inductive arguments on path lengths.

ABSTRACT

The bondage number $b(G)$ of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with domination number greater than that of $G$. Denote $P_n imes P_m$ be the Cartesian product of two paths $P_n$ and $P_m$. This paper determines that the exact value of $b(P_n imes P_2)$, $b(P_n imes P_3)$ and $b(P_n imes P_4)$ for $n\ge 2$.

Motivation & Objective

• To determine the exact bondage number of mesh networks $P_n \times P_m$ for small values of $m$.
• To analyze the robustness of mesh networks under edge failures by measuring the minimum number of edges whose removal increases the domination number.
• To extend prior results on bondage numbers in special graph classes to Cartesian products of paths.
• To propose a conjecture on the bondage number of larger mesh networks based on observed patterns.

Proposed method

• The authors use structural analysis of dominating sets in $P_n \times P_m$ graphs, focusing on vertex and edge configurations in subgraphs.
• They apply inductive reasoning on the path length $n$, assuming results hold for smaller $n$ and proving them for larger $n$.
• The method involves case analysis based on the number of dominating set vertices in the first few rows of the mesh.
• They leverage known results on domination numbers of smaller subgraphs ($H_{k,4}$) to bound the size of dominating sets in larger graphs.
• The proof relies on contradiction: assuming a minimum dominating set exists after edge removal, and showing it must be larger than the original domination number.
• They use lemmas to establish that certain configurations of dominating sets cannot be minimal, thus proving the bondage number is at least a given value.

Experimental results

Research questions

• RQ1 What is the exact value of the bondage number $b(P_n \times P_2)$ for $n \geq 2$?
• RQ2 What is the exact value of the bondage number $b(P_n \times P_3)$ for $n \geq 4$?
• RQ3 What is the exact value of the bondage number $b(P_n \times P_4)$ for $n \geq 4$, excluding specific cases?
• RQ4 How does the structure of the mesh network affect the minimum number of edge failures that can disrupt all minimum dominating sets?
• RQ5 Can a general bound on the bondage number be established for larger mesh networks ($m \geq 5$)?

Key findings

• The bondage number $b(P_n \times P_2) = 1$ for all $n \geq 2$, meaning removing a single edge can increase the domination number.
• The bondage number $b(P_n \times P_3) = 1$ for $n \geq 4$, indicating that a single edge removal can disrupt all minimum dominating sets.
• For $P_n \times P_4$, the bondage number is 1 for all $n \geq 4$ except $n = 5, 6, 9$, where it is 3, 2, and 3 respectively.
• The domination number of $G_{n,4}$ remains unchanged after removing vertex $u_{1,1}$, suggesting structural resilience in corner vertices.
• The authors establish that $b(G_{n,4}) = 1$ for $n \notin \{5,6,9\}$, with $n \geq 4$, using inductive and case-based arguments.
• A conjecture is proposed that $b(G_{n,m}) \leq 2$ for all $m \geq 5$, suggesting bounded robustness for larger mesh networks.