Directed Steiner tree packing and directed tree connectivity

Abstract

AbstractFor a digraph , and a set with and , an ‐tree is an out‐tree rooted at with . Two ‐trees and are said to be arc‐disjoint if . Two arc‐disjoint ‐trees and are said to be internally disjoint if . Let and be the maximum number of internally disjoint and arc‐disjoint ‐trees in , respectively. The generalized ‐vertex‐strong connectivity of is defined as Similarly, the generalized ‐arc‐strong connectivity of is defined as The generalized ‐vertex‐strong connectivity and generalized ‐arc‐strong connectivity are also called directed tree connectivity which extends the well‐established tree connectivity on undirected graphs to directed graphs and could be seen as a generalization of classical connectivity of digraphs. In this paper, we completely determine the complexity for both and on general digraphs, symmetric digraphs, and Eulerian digraphs. In particular, among our results, we prove and use the NP‐completeness of 2‐linkage problem restricted to Eulerian digraphs. We also give sharp bounds and equalities for the two parameters and .