1-Extra 3-component edge connectivity of modified bubble-sort networks

The classic edge connectivity of a network G is the minimum number of edges whose removal disconnects G into multiple components. It can be used as a measure of the reliability of the network. The extra edge connectivity (resp. the component edge connectivity) imposes conditions on the size (resp. the number) of components, and both are the generalizations of the classic edge connectivity. The extra component edge connectivity combines both conditional edge connectivities to measure the network’s reliability in a more particular and targeted manner. The modified bubble-sort graph \(MB_n\) $M{B}_n$ is a topological model proposed for the interconnection network of multicomputer systems. In this paper, we study the 1-extra 3-component edge connectivity \(c\lambda _3^1 (MB_n )\) $c{\lambda }_3^1(M{B}_n)$ of the modified bubble-sort graph and prove that \(c\lambda _3^1 (MB_n )=4n-6\) $c{\lambda }_3^1(M{B}_n)=4n-6$ for \(n\ge 7\) $n\ge 7$ .