Perfect Out-forests and Steiner Cycle Packing in Digraphs

Abstract

In this paper, we study the complexity of two types of digraph packing problems: perfect out-forests problem and Steiner cycle packing problem.For the perfect out-forest problem, we prove that it is NP-hard to decide whether a given strong digraph contains a 1-perfect outforest. However, when restricted to a semicomplete digraph D, the problem of deciding whether D contains an i-perfect out-forest becomes polynomial-time solvable, where i ∈ {0, 1}. We also prove that it is NP-hard to find a 0-perfect out-forest of maximum size in a connected acyclic digraph, and it is NP-hard to find a 1-perfect out-forest of maximum size in a connected digraph.For the Steiner cycle packing problem, when both k ≥ 2, ℓ ≥ 1 are fixed integers, we show that the problem of deciding whether there are at least ℓ internally disjoint directed S-Steiner cycles in an Eulerian digraph D is NP-complete, where S ⊆ V (D) and |S| = k. However, when we consider the class of symmetric digraphs, the problem becomes polynomial-time solvable. We also show that the problem of deciding whether there are at least ℓ arc-disjoint directed S-Steiner cycles in a given digraph D is NP-complete, where S ⊆ V (D) and |S| = k.