A Lower Bound for the 3-Pendant Tree-Connectivity of Lexicographic Product Graphs

For a connected graph G = (V, E) and a set S subset of 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 subset of 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 subset of V(G) and | S| >= 2, the pendant tree-connectivity tau(G)(S) is the maximum number of internally disjoint pendant S-Steiner trees in G, and for k >= 2, the k-pendant tree-connectivity tau(k)(G) is the minimum value of tau(G)(S) over all sets S of k vertices. We derive a lower bound for tau(3)(G o H), where G and H are connected graphs and o denotes the lexicographic product.