Packing the Steiner trees of a graph

Let G = (V,E) be an undirected graph with a distinguished set of terminal vertices K ⊆ V, |K| ≥ 2. A K-Steiner tree T of G is a tree containing the terminal vertex-set K, where any vertex of degree one in T must belong to K. The Steiner Tree Packing problem (STPP for short) is the problem of finding the maximum number of edge-disjoint K-Steiner trees, t_K(G), contained in G. Specifically we are interested in finding a lower bound on t_K(G) with respect to the K-edge-connectivity, denoted as λ_K(G). In 2003, Kriesell conjectured that any graph G with terminal vertex-set K has at least ⌊λ_K(G)-2⌋ edge-disjoint K-Steiner trees. In this article, we show that this conjecture can be answered affirmatively if the edges of G can be partitioned into K-Steiner trees. This result yields bounds for the problem of packing K-Steiner trees with certain intersection properties in a graph. In addition we show that for any graph G with terminal vertex-set K, t_K(G) ≥ ⌊λ_K(G)-2⌋ - |V - K|-2 - 1. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009