The 3-Path Connectivity of Dragonfly Networks

Dragonfly networks $D(n,h)$ are a class of interconnection topologies widely used for large-scale high-performance computing (HPC) systems. In such networks, path connectivity serves as a fundamental metric for evaluating fault tolerance and operational reliability. Let G be a connected simple graph with vertex set $V\left(G\right)$. Let $\Omega$ be a subset of $V\left(G\right)$ with cardinality at least two. A path containing all vertices of $\Omega$ is said to be an $\Omega$-path of G. Two paths ($T_1$ and $T_2$) of G are internally disjoint if $V\left(T_1\right)\cap V\left(T_2\right)=\Omega$ and $E\left(T_1\right)\cap E\left(T_2\right)=\varnothing$. For an integer with $2\le \ell$, the ℓ-path connectivity ${\pi }_{\ell }\left(G\right)$ is defined as ${\pi }_{\ell }\left(G\right)=min\{{\pi }_G\left(\Omega \right)|\Omega \subseteq V\left(G\right)\mathrm{and}|\Omega |=\ell \}$, where ${\pi }_G\left(\Omega \right)$ represents the maximum number of internally disjoint $\Omega$-paths. This paper focuses on resolving the exact value of 3-path connectivity of dragonfly networks, ${\pi }_3\left(D(n,h)\right)$, defined as the maximum number of internally disjoint paths among any three distinct vertices in $D(n,h)$. For $D(n,h)$ with $n\ge 5$ and $h\ge 2$, the exact 3-path connectivity is ${\pi }_3\left(D(n,h)\right)=\lfloor {\displaystyle \frac{3h+2n}{4}}\rfloor$ if $h\le n-2$, and ${\pi }_3\left(D(n,h)\right)=\lfloor {\displaystyle \frac{3n+2h-2}{4}}\rfloor$ if $h\ge n-1$.