Reliability analysis of twisted cubes

Connectivity is a vital metric to explore fault tolerance and reliability of network structure based on a graph model. Let G=(V,E) be a connected graph. A connected graph G is called supper-κ (resp. super-λ) if every minimum vertex cut (edge cut) of G is the set of neighbors of some vertex in G. Let F⊆V be a vertex set, F is called extra-cut, if G−F is not connected and each component of G−F has more than k vertices. The extraconnectivity κk(G) is the cardinality of the minimum extra-cuts. A r-component cut of G is a set S of vertices, G−S has at least r components. r-component connectivity cκr(G) of G is the size of the smallest r-component cut. The r-component edge connectivity cλr(G) can be defined similarly. In this paper, we determine the r-component (edge) connectivity of twisted cubes TNn for small r. And we also prove other properties of TNn.