The globally bi-3* and hyper bi-3* connectedness of the spider web networks

Abstract Assume that m and n are positive even integers with m  ⩾ 4. The spider web network SW ( m ,  n ) is recognized as a variation of the honeycomb rectangular mesh HREM ( m ,  n ) [I. Stojmenovic, Honeycomb Networks: Topological Properties and Communication algorithms, IEEE Trans. Parallel and Distributed Systems 8 (1997) 1036–1042. [10] ], and is a 3-regular bipartite planar graph. Suppose that SW ( m ,  n ) has bipartition V 1 and V 2 . In this paper, we prove that for any x  ∈  V 1 and y  ∈  V 2 , there exist three internally disjoint paths between x and y whose union spans SW ( m ,  n ). Moreover, for any three vertices x ,  y and z of the same partite set, there exist three internally disjoint paths between x and y whose union spans SW ( m ,  n ) −  z . Such results imply that SW ( m ,  n ) remains hamiltonian when it contains a pair of faulty vertices of the opposite partite sets, or when it contains a faulty edge. More precisely, SW ( m ,  n ) − { x ,  y } is hamiltonian for any x  ∈  V 1 and y  ∈  V 2 , and SW ( m ,  n ) −  e is hamiltonian for any e  ∈  E ( SW ( m ,  n )).