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 subset of 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 subset of 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) boolean AND E(T') = empty set and V (T) boolean AND V (T') = S. For S subset of V (G) and vertical bar S vertical bar >= 2, the local pendant tree-connectivity tau(G)(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 tau(k)(G) = min{tau(G)(S) vertical bar S subset of V (G), vertical bar S vertical bar = k}. In this paper, we prove that for any two connected graphs G and H, tau(3)(G square H) >= min{3left perpendicular tau(3)(G)/2right perpendicular, 3left perpendicular tau(3)(H)/2right perpendicular}. Moreover, the bound is sharp.