Link fault tolerance of BC networks and folded hypercubes on h-extra r-component edge-connectivity

Parallel and distributed systems play a significant role in high-performance computing, prompting us to investigate qualitative and quantitative metrics to indicate the fault tolerance and vulnerability of systems. Consider the setup where there are large-scale link malfunctions that disconnect the network and result in various components, each with multiple processors. In this paper, we propose and study the h-extra r-component edge-connectivity of a connected graph G, which is denoted by cλrh(G) and has not been addressed before. Let h and r be two positive integers with r≥2. An edge subset F⊆E(G) is said to be an h-extra r-component edge-cut of G, if any, G−F has at least r components and every component of G−F has at least h vertices. The cardinality of the minimum h-extra r-component edge-cut of G is the h-extra r-component edge-connectivity of G. Let h be a positive integer with the decomposition h=∑i=0t2ki, where ki>ki+1, 0≤i≤t−1. In this paper, we derive a lower bound for the exact value of h-extra 3-component edge-connectivity of BC networks Bn and demonstrate that it is tight for one member of Bn, hypercube Qn, in the interval 1≤h≤2⌊n2⌋−1−1, n≥4. This lower bound is also tight for the exact value of 2c-extra 3-component edge-connectivity of Bn with 0≤c≤n−2, n≥4. Specifically, cλ3h(Qn)=2nh−∑i=0tki2ki+1−∑i=0ti2ki+2−h for 1≤h≤2⌊n2⌋−1−1, n≥4 and cλ32c(Bn)=(2n−2c−1)2c for 0≤c≤n−2, n≥4. Exact value of h-extra 3-component edge-connectivity of n-dimensional folded hypercube FQn, cλ3h(FQn) is 2(n+1)h−∑i=0tki2ki+1−∑i=0ti2ki+2−h for 1≤h≤2⌈n2⌉−1−1, n≥4, and that of 2c-extra 3-component edge-connectivity of FQn, cλ32c(FQn), is (2n−2c+1)2c for 0≤c≤n−2, n≥4.