Packing internally disjoint Steiner paths of modified bubble-sort networks

Abstract

Let G be a connected simple graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. For $S\subseteq V\left(G\right)$ with $\left|S\right|\ge 2$, a path P in G is said to be an S-Steiner path (or S-path for short) if it connects all vertices of S. Two S-paths $P_1$ and $P_2$ are internally disjoint if $E\left(P_1\right)\cap E\left(P_2\right)=\varnothing$ and $V\left(P_1\right)\cap V\left(P_2\right)=S$. The packing number of internally disjoint S-paths, denoted as ${\pi }_G\left(S\right)$, is the maximum number of internally disjoint S-paths in G. For an integer $2\le k\le \left|V\right(G\left)\right|$, the k-path connectivity of a graph G is defined as ${\pi }_k\left(G\right)=$ min$\left\{{\pi }_G\right(S\left)\right|S\subseteq V\left(G\right)$ and $\left|S\right|=k\}$. The modified bubble-sort graph, denoted $M{B}_n$, is an interconnection network topological model for multiprocessor systems. In this paper, we focus on the 3-path-connectivity of the n-dimensional modified bubble sort graphs. By analyzing the structural properties of $M{B}_n$, we determine that ${\pi }_3\left(M{B}_n\right)=\lfloor \frac{3n-1}{4}\rfloor$ for $n\ge 4$.