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) 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 F ⊆ V be a vertex set, F is called extra-cut, if G−F G − F is not connected and each component of G−F G − F has more than k vertices. The extraconnectivity κ k (G) κ k ( G ) is the cardinality of the minimum extra-cuts. A r -component cut of G is a set S of vertices, G−S G − S has at least r components. r -component connectivity cκ r (G) c κ r ( G ) of G is the size of the smallest r -component cut. The r -component edge connectivity cλ r (G) c λ r ( G ) can be defined similarly. In this paper, we determine the r -component (edge) connectivity of twisted cubes TN n T N n for small r . And we also prove other properties of TN n T N n .