Packing internally disjoint Steiner trees to compute the κ3-connectivity in augmented cubes

Abstract

Given a connected graph G and S⊆V(G) with |S|≥2, a tree T in G is called an S-Steiner tree (or S-tree for short) if S⊆V(T). Two S-trees T1 and T2 are internally disjoint if E(T1)∩E(T2)=∅ and V(T1)∩V(T2)=S. The packing number of internally disjoint S-trees, denoted as κG(S), is the maximum size of a set of internally disjoint S-trees in G. For an integer k≥2, the generalized k-connectivity (abbr. κk-connectivity) of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}. The n-dimensional augmented cube, denoted as AQn, is an important variant of the hypercube that possesses several desired topology properties such as diverse embedding schemes in applications of parallel computing. In this paper, we focus on the study of constructing internally disjoint S-trees with |S|=3 in AQn. As a result, we completely determine the κ3-connectivity of AQn as follows: κ3(AQ4)=5 and κ3(AQn)=2n−2 for n=3 or n≥5.