[Paper Review] Component edge connectivity of the folded hypercube

Shu-Li Zhao, Weihua Yang|arXiv (Cornell University)|Mar 4, 2018

Interconnection Networks and Systems12 references3 citations

TL;DR

This paper determines the (g+1)-component edge connectivity of the folded hypercube $FQ_n$ for $n \geq 5$ and $g \leq 2^{\lfloor (n+1)/2 \rfloor}$, establishing a closed-form formula based on the binary decomposition of $g$. The key result is $c\lambda_{g+1}(FQ_n) = (n+1)g - \left(\sum_{i=0}^{s} t_i 2^{t_i - 1} + \sum_{i=0}^{s} i \cdot 2^{t_i}\right)$, where $g = \sum_{i=0}^{s} 2^{t_i}$, providing a precise measure of fault tolerance in this interconnection network.

ABSTRACT

The $g$-component edge connectivity $cλ_g(G)$ of a non-complete graph $G$ is the minimum number of edges whose deletion results in a graph with at least $g$ components. In this paper, we determine the component edge connectivity of the folded hypercube $cλ_{g+1}(FQ_{n})=(n+1)g-(\sum\limits_{i=0}^{s}t_i2^{t_i-1}+\sum\limits_{i=0}^{s} i\cdot 2^{t_i})$ for $g\leq 2^{[\frac{n+1}2]}$ and $n\geq 5$, where $g$ be a positive integer and $g=\sum\limits_{i=0}^{s}2^{t_i}$ be the decomposition of $g$ such that $t_0=[\log_{2}{g}],$ and $t_i=[\log_2({g-\sum\limits_{r=0}^{i-1}2^{t_r}})]$ for $i\geq 1$.

Motivation & Objective

• To determine the $g$-component edge connectivity $c\lambda_{g+1}(FQ_n)$ of the folded hypercube $FQ_n$ for $n \geq 5$ and $g \leq 2^{\lfloor (n+1)/2 \rfloor}$.
• To extend the concept of standard edge connectivity to component edge connectivity, offering a more refined measure of fault tolerance in interconnection networks.
• To provide a precise analytical formula for $c\lambda_{g+1}(FQ_n)$ using the binary decomposition of $g$.
• To establish that the minimum edge cut disconnecting $FQ_n$ into at least $g+1$ components is achieved by specific subgraph structures, particularly incomplete folded hypercubes.

Proposed method

• The authors define $g$ as a sum of powers of two, $g = \sum_{i=0}^{s} 2^{t_i}$, with $t_0 = \lfloor \log_2 g \rfloor$ and subsequent $t_i$ recursively defined.
• They utilize the maximum number of edges in induced subgraphs of $FQ_n$, denoted $ex_m(FQ_n)$, derived from known results on hypercubes and extended to folded hypercubes.
• The proof employs structural decomposition of $FQ_n$ as $D_0 \otimes D_1$, where $D_0$ and $D_1$ are $(n-1)$-dimensional subcubes, to analyze component structures after edge deletion.
• The method involves constructing a minimal edge cut $F$ that disconnects $FQ_n$ into $g+1$ components, with $g$ isolated vertices and one larger component, and proving $|F| = (n+1)g - \frac{1}{2}ex_g$.
• The authors use inequalities and subgraph edge maximization techniques, particularly leveraging the properties of incomplete folded hypercubes $LF_m$ and reverse incomplete folded hypercubes $RF_m$, to bound the number of edges in component-inducing subgraphs.
• They apply induction and case analysis based on the size of the component containing more than one vertex, distinguishing between cases where this component has fewer than $2^{n-2}$ vertices and otherwise.

Experimental results

Research questions

• RQ1 What is the exact value of the $(g+1)$-component edge connectivity $c\lambda_{g+1}(FQ_n)$ for the folded hypercube $FQ_n$ when $n \geq 5$ and $g \leq 2^{\lfloor (n+1)/2 \rfloor}$?
• RQ2 How does the component edge connectivity of $FQ_n$ depend on the binary decomposition of $g$?
• RQ3 Can the minimum edge cut that produces $g+1$ components in $FQ_n$ be characterized by specific subgraph structures such as incomplete folded hypercubes?
• RQ4 Is the formula $c\lambda_{g+1}(FQ_n) = (n+1)g - \frac{1}{2}ex_g$ valid for all $g \leq 2^{\lfloor (n+1)/2 \rfloor}$ and $n \geq 5$?

Key findings

• The $(g+1)$-component edge connectivity of the folded hypercube $FQ_n$ is exactly $c\lambda_{g+1}(FQ_n) = (n+1)g - \left(\sum_{i=0}^{s} t_i 2^{t_i - 1} + \sum_{i=0}^{s} i \cdot 2^{t_i}\right)$ for $n \geq 5$ and $g \leq 2^{\lfloor (n+1)/2 \rfloor}$.
• The minimum edge cut that disconnects $FQ_n$ into $g+1$ components consists of $g$ isolated vertices and one remaining component, confirming that the optimal cut isolates $g$ vertices.
• The value $ex_g$ represents the maximum number of edges in any induced subgraph of $FQ_n$ on $g$ vertices, and this maximum is achieved by the incomplete folded hypercube $LF_g$.
• The formula is derived by analyzing edge cuts in $FQ_n$ decomposed into two $(n-1)$-dimensional subcubes $D_0$ and $D_1$, with careful counting of edges incident to components.
• The proof holds for both even and odd $n$, with the case $|V(C_{g+1})| \geq 2^{n-2}$ handled via subgraph edge maximization and inequality chains involving $ex_m$ and $ex_{m-g+1}$.
• The result is not complete for $g > 2^{\lfloor (n+1)/2 \rfloor} + 1$, leaving the problem open for larger $g$.