Maximally Local Connectivity on Augmented Cubes

Abstract

Connectivity is an important measurement for the fault tolerance in interconnection networks. It is known that the augmented cube AQ _n is maximally connected, i.e. (2n - 1)-connected, for n ≥ 4. By the classical Menger’s Theorem, every pair of vertices in AQ _n is connected by 2n - 1 vertex-disjoint paths for n ≥ 4. A routing with parallel paths can speed up transfers of large amounts of data and increase fault tolerance. Motivated by some research works on networks with faults, we have a further result that for any faulty vertex set F ⊂ V(AQ _n ) and |F| ≤ 2n − 7 for n ≥ 4, each pair of non-faulty vertices, denoted by u and v, in AQ _n  − F is connected by min{deg _f (u), deg _f (v)} vertex-disjoint fault-free paths, where deg _f (u) and deg _f (v) are the degree of u and v in AQ _n  − F, respectively. Moreover, we have another result that for any faulty vertex set F ⊂ V(AQ _n ) and |F| ≤ 4n − 9 for n ≥ 4, there exists a large connected component with at least 2ⁿ − |F| − 1 vertices in AQ _n  − F. In general, a remaining large fault-free connected component also increases fault tolerance.

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 96-2221-E-009-137-MY3. This research was partially supported by the Aiming for the Top University and Elite Research Center Development Plan.