[Paper Review] Generalized Measures of Fault Tolerance in (n,k)-star Graphs

Xiang-Jun Li, Jun‐Ming Xu|arXiv (Cornell University)|Apr 6, 2012

Interconnection Networks and Systems6 references3 citations

TL;DR

This paper establishes the h-super connectivity of (n,k)-star graphs, proving that κₛ⁽ʰ⁾(Sₙ,ₖ) = n + h(k−2)−1 for 2 ≤ k ≤ n−1 and 0 ≤ h ≤ n−k. The result generalizes prior fault tolerance measures, providing a precise threshold for vertex removal that disconnects the graph while preserving minimum degree h, using structural decomposition and inductive arguments on subgraphs.

ABSTRACT

This paper considers a kind of generalized measure $κ_s^{(h)}$ of fault tolerance in the $(n,k)$-star graph $S_{n,k}$ and determines $κ_s^{(h)}(S_{n,k})=n+h(k-2)-1$ for $2 \leqslant k \leqslant n-1$ and $0\leqslant h \leqslant n-k$, which implies that at least $n+h(k-2)-1$ vertices of $S_{n,k}$ have to remove to get a disconnected graph that contains no vertices of degree less than $h$. This result contains some known results such as Yang et al. [Information Processing Letters, 110 (2010), 1007-1011].

Motivation & Objective

• To generalize fault tolerance measures in (n,k)-star graphs beyond classical vertex connectivity.
• To determine the h-super connectivity κₛ⁽ʰ⁾(G) for (n,k)-star graphs Sₙ,ₖ, which measures the minimum vertex cut ensuring remaining graph has minimum degree h.
• To unify and extend known results on κₛ⁽¹⁾(Sₙ) and κₛ⁽²⁾(Sₙ) for star graphs.
• To provide a precise analytical formula for fault tolerance under higher-degree resilience constraints.

Proposed method

• Uses structural decomposition of Sₙ,ₖ into subgraphs Sⁿ⁻¹ₖ₋₁ via fixed bit positions, exploiting isomorphism to Sₙ₋₁,ₖ₋₁.
• Applies inductive reasoning on subgraphs, analyzing vertex cuts based on partitioning vertices into sets Xᵢ and Yᵢ relative to a fixed bit position.
• Employs lemmas on complete subgraphs Kₙ₋ₖ₊₁ induced by fixed (k−1)-permutations and independent swap-edges between subgraphs.
• Analyzes three cases based on the size of the index set J′ (vertices with h−1 neighbors in the cut), using bounds from (n−2)!/(n−k)! for swap-edge counts.
• Applies known results for k=n−1 (star graphs) as base cases, verifying consistency with prior work.
• Uses contradiction and extremal arguments to bound minimum cut size, proving the formula holds under all parameter constraints.

Experimental results

Research questions

• RQ1 What is the h-super connectivity of the (n,k)-star graph Sₙ,ₖ for general h and k?
• RQ2 How does the h-super connectivity generalize classical connectivity and 1-super connectivity in (n,k)-star graphs?
• RQ3 Can the fault tolerance of (n,k)-star graphs be precisely quantified when the remaining graph must maintain minimum degree h?
• RQ4 Does the formula κₛ⁽ʰ⁾(Sₙ,ₖ) = n + h(k−2)−1 hold across the full parameter range 2 ≤ k ≤ n−1 and 0 ≤ h ≤ n−k?

Key findings

• The h-super connectivity of Sₙ,ₖ is exactly n + h(k−2)−1 for 2 ≤ k ≤ n−1 and 0 ≤ h ≤ n−k.
• This result generalizes Yang et al.'s earlier findings on κₛ⁽¹⁾(Sₙ,ₖ) = n+k−3 and κₛ⁽²⁾(Sₙ,ₖ) = n+2k−5.
• For k=n−1, the formula reduces to κₛ⁽¹⁾(Sₙ) = 2n−4, consistent with known results on star graphs.
• The bound is tight, as shown by constructing vertex cuts of size n + h(k−2)−1 that disconnect the graph while preserving minimum degree h.
• The result is invalid for h ≥ 3 in the standard star graph Sₙ, indicating limitations in extending the formula to higher h.
• The proof relies on inductive decomposition and edge-counting arguments on subgraphs Sⁿ⁻¹ₖ₋₁, confirming the formula across all valid parameter ranges.