Matching preclusion and strong matching preclusion of the bubble-sort star graphs

TL;DR

This paper investigates the robustness of bubble-sort star graphs in distributed systems by analyzing their matching preclusion numbers, revealing specific vulnerabilities and optimal fault sets.

Contribution

It determines the strong and regular matching preclusion numbers of bubble-sort star graphs and characterizes their optimal fault sets, advancing understanding of their fault tolerance.

Findings

Strong matching preclusion number of $BS_n$ is 2 for $n extgreater=3$.

Matching preclusion number of $BS_n$ is $2n-3$ for $n extgreater=3$.

Optimal preclusion sets are characterized as specific vertex sets.

Abstract

Since a plurality of processors in a distributed computer system working in parallel, to ensure the fault tolerance and stability of the network is an important issue in distributed systems. As the topology of the distributed network can be modeled as a graph, the (strong) matching preclusion in graph theory can be used as a robustness measure for missing edges in parallel and distributed networks, which is defined as the minimum number of (vertices and) edges whose deletion results in the remaining network that has neither a perfect matching nor an almost-perfect matching. The bubble-sort star graph is one of the validly discussed interconnection networks related to the distributed systems. In this paper, we show that the strong matching preclusion number of an $n$-dimensional bubble-sort star graph $BS_n$ is $2$ for $n\geq3$ and each optimal strong matching preclusion set of $BS_n$ is a set of two vertices from the same bipartition set. Moreover, we show that the matching preclusion number of $BS_n$ is $2n-3$ for $n\geq3$ and that every optimal matching preclusion set of $BS_n$ is trivial.