Pendant 3-tree Connectivity of Augmented Cubes

Abstract

The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set S of vertices, |S| ≥ 2, a tree connecting all vertices of S is called an S-Steiner tree (tree connecting S). The reliability of a network G to connect any S vertices (|S| number of vertices) in G can be measure by this parameter. For an S-Steiner tree, if the degree of each vertex in S is equal to one, then that tree is called a pendant S-Steiner tree. Two pendant S-Steiner trees T and T ′ are said to be internally disjoint ifThe local pendant tree-connectivity τG(S) is the maximum number of internally disjoint pendant S-Steiner trees in G. For an integer k with 2 ≤ k ≤ n, the pendant k-tree-connectivity is defined as τ k (G) = min{τG(S) : S ⊆ V (G), |S| = k}. In this paper, we study the pendant 3tree connectivity of Augmented cubes which are modification of hypercubes invented to increase the connectivity and decrease the diameter hence superior to hypercubes. We show that τ3(AQn) = 2n − 3. , which attains the upper bound of τ3(G) given by Hager, for G = AQn.