Fault-tolerability analysis of hypercubes based on 3-component path-structure connectivity

Interconnection networks are essential in parallel computing and network science nowadays. Network failures are inevitable during operation and result in inestimable losses. Hence, designing an interconnection network with excellent performance is necessary. Reliability is a key indicator of the performance of interconnection networks, and its research originated from the first telecommunication switching network system. The failure of elements in a network system reduces overall communication capacity, leading to network congestion and system failure, typically measured by connectivity. In this paper, we introduce a new type of conditional connectivity of a graph $G$, termed $r$-component $H$-structure connectivity and denoted as $c{\kappa }_r(G;H)$. Then, we investigate 3-component $P_k$-structure connectivity for hypercube networks $Q_n$ and acquire the result $c{\kappa }_3(Q_n;P_k)=\left(\begin{array}{cc}2n-4& \text{if}k=2;\\ \lceil \frac{4n-5}{k}\rceil & \text{for}k\ge 4\text{even};\\ \lceil \frac{4n-5}{k+1}\rceil & \text{for}k\ge 3\text{odd}.\end{array}\right)$