Edge-fault-tolerant strong Menger edge connectivity of bubble-sort star graphs

TL;DR

This paper investigates the fault tolerance of the n-dimensional bubble-sort star graph, proving its strong Menger edge connectivity resilience under multiple edge failures, with optimal bounds established.

Contribution

It establishes the exact edge-fault-tolerant strong Menger edge connectivity levels of bubble-sort star graphs, a novel analysis for this network topology.

Findings

BS_n is (2n-5)-edge-fault-tolerant strongly Menger edge connected for n≥3.

BS_n is (6n-17)-conditional edge-fault-tolerant strongly Menger edge connected for n≥4.

Results are proven to be optimal through examples.

Abstract

The connectivity and edge connectivity of interconnection network determine the fault tolerance of the network. An interconnection network is usually viewed as a connected graph, where vertex corresponds processor and edge corresponds link between two distinct processors. Given a connected graph $G$ with vertex set $V(G)$ and edge set $E(G)$, if for any two distinct vertices $u,v\in V(G)$, there exist $\min\{d_G(u),d_G(v)\}$ edge-disjoint paths between $u$ and $v$, then $G$ is strongly Menger edge connected. Let $m$ be an integer with $m\geq1$. If $G-F_e$ remains strongly Menger edge connected for any $F_e\subseteq E(G)$ with $|F_e|\leq m$, then $G$ is $m$-edge-fault-tolerant strongly Menger edge connected. If $G-F_e$ is strongly Menger edge connected for any $F_e\subseteq E(G)$ with $|F_e|\leq m$ and $\delta(G-F_e)\geq2$, then $G$ is $m$-conditional edge-fault-tolerant strongly Menger edge connected. In this paper, we consider the $n$-dimensional bubble-sort star graph $BS_n$. We show that $BS_n$ is $(2n-5)$-edge-fault-tolerant strongly Menger edge connected for $n\geq3$ and $(6n-17)$-conditional edge-fault-tolerant strongly Menger edge connected for $n\geq4$. Moreover, we give some examples to show that our results are optimal.