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 τG(S) is the maximum number of internally disjoint pendant S-Steiner trees in G, and for k ≽ 2, the k-pendant tree-connectivity τk(G) is the minimum value of τG(S) over all sets S of k vertices. We derive a lower bound for τ3(G ◦ H), where G and H are connected graphs and ◦ denotes the lexicographic product.