Pendant 3-tree-connectivity of augmented cubes

The Steiner tree problem in graphs is widely studied because of its usefulness in network design and circuit layout. In this context, given a set of vertices \(S(|S| \ge 2,)\) $S(|S|\ge 2,)$ a tree that connects all vertices in S is called an S-Steiner tree. This helps to measure how well a network G can connect any set of S vertices together. In an S-Steiner tree, if each vertex in S has only one connection, it is called a pendant S-Steiner tree. Two pendant S-Steiner trees, T and \(T',\) $T^{\prime },$ are internally disjoint if \(E(T) \cap E(T') = \emptyset\) $E(T)\cap E(T^{\prime })=\mathrm{\varnothing }$ and \(V(T) \cap V(T') = S.\) $V(T)\cap V(T^{\prime })=S.$ The local pendant tree-connectivity, denoted as \(\tau _{G}(S),\) ${\tau }_G(S),$ represents the maximum number of internally disjoint pendant S-Steiner trees in graph G. For an integer k with \(2 \le k \le n,\) $2\le k\le n,$ where n is the number of vertices, the pendant k-tree-connectivity, denoted as \(\tau _{k}(G),\) ${\tau }_k(G),$ is defined as \(\tau _{k}(G) = min\{ \tau _{G}(S): S \subseteq V(G), |S| = k\}.\) ${\tau }_k(G)=min\{{\tau }_G(S):S\subseteq V(G),|S|=k\}.$ This paper focuses on studying the pendant 3-tree-connectivity of augmented cubes, which are modified versions of hypercubes designed to enhance connectivity and reduce diameter. This research demonstrates that the pendant 3-tree-connectivity of augmented cubes, denoted as \(\tau _3(AQ_n)\) ${\tau }_3(A{Q}_n)$ is \(2n-3\) $2n-3$ . This result matches the upper bound of \(\tau _3(G)\) ${\tau }_3(G)$ provided by Hager, specifically for the augmented cube graph \(AQ_n\) $A{Q}_n$ .