Constructing internally disjoint pendant Steiner trees in Cartesian product networks

The concept of pendant tree-connectivity was introduced by Hager in 1985. For a graph G = (V,E) and a set S ⊆ V (G) of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a subgraph T = (V ′, E ′) of G that is a tree with S ⊆ V ′. For an S-Steiner tree, if the degree of each vertex in S is equal to one, then this 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 )∩E(T ′) = ∅ and V (T ) ∩ V (T ′) = S. For S ⊆ V (G) and |S| ≥ 2, the local pendant treeconnectivity τG(S) is the maximum number of internally disjoint pendant S-Steiner trees in G. For an integer k with 2 ≤ k ≤ n, pendant tree k-connectivity is defined as τk(G) = min{τG(S) |S ⊆ V (G), |S| = k}. In this paper, we prove that for any two connected graphs G and H , τ3(G H) ≥ min{3 τ3(G) 2 , 3 τ3(H) 2 }. Moreover, the bound is sharp.