Pendant 3-tree Connectivity of Augmented Cubes

The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set \(S\) of vertices, \(|S| \geq 2,\) 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 if \(E(T) \cap E(T') = \emptyset\) and \(V(T) \cap V(T') = S.\) The local pendant tree-connectivity \(\tau_{G}(S)\) is the maximum number of internally disjoint pendant \(S\)-Steiner trees in \(G.\) For an integer \(k\) with \(2 \leq k \leq n,\) the pendant k-tree-connectivity is defined as \(\tau_{k}(G) = min\{ \tau_{G}(S) : S \subseteq V(G), |S| = k\}.\) In this paper, we study the pendant \(3\)-tree connectivity of Augmented cubes which are modifications of hypercubes invented to increase the connectivity and decrease the diameter hence superior to hypercubes. We show that \(\tau_3(AQ_n) = 2n-3.\) , which attains the upper bound of \(\tau_3(G)\) given by Hager, for \(G = AQ_n\).