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 | ≥ 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 ′ , are internally disjoint if E (T) ∩ E (T ′) = ∅ and V (T) ∩ V (T ′) = S. The local pendant tree-connectivity, denoted as τ G (S) , represents the maximum number of internally disjoint pendant S-Steiner trees in graph G. For an integer k with 2 ≤ k ≤ n , where n is the number of vertices, the pendant k-tree-connectivity, denoted as τ k (G) , is defined as τ k (G) = m i n { τ G (S) : S ⊆ 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 τ 3 (A Q n) is 2 n - 3 . This result matches the upper bound of τ 3 (G) provided by Hager, specifically for the augmented cube graph A Q n . [ABSTRACT FROM AUTHOR]