A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

for a connected graph G = ( V, E ) and a set S ⊆ V ( G ) with at least two vertices, an S -Steiner tree is a subgraph T = ( V ′, E ′) of G that is a tree with S ⊆ V ′. If the degree of each vertex of S in T is equal to 1, then T is called a pendant S -Steiner tree. Two S -Steiner trees are internally disjoint if they share no vertices other than S and have no edges in common. For S ⊆ V ( G ) and | S | ≽ 2, the pendant tree-connectivity τ G ( S ) is the maximum number of internally disjoint pendant S -Steiner trees in G , and for k ≽ 2, the k -pendant tree-connectivity τ k ( G ) is the minimum value of τ G ( S ) over all sets S of k vertices. We derive a lower bound for τ 3 ( G ◦ H ), where G and H are connected graphs and ◦ denotes the lexicographic product.