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, $|S| \geq 2,$ 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 $G = AQ_n$.