Oriented star packings

Abstract

Given a (possibly infinite) family S of oriented stars, an S -packing in a digraph D is a collection of vertex disjoint subgraphs of D, each isomorphic to a member of S . The S - Packing problem asks for the maximum number of vertices, of a host digraph D, that can be covered by an S -packing of D. We prove a dichotomy for the decision version of the S -packing problem, giving an exact classification of which problems are polynomial time solvable and which are NP-complete. For the polynomial problems, we provide Hall type min-max theorems, including versions for (locally) degree-constrained variants of the problems. An oriented star can be specified by a pair of ( k , ℓ ) ∈ N 2 ∖ ( 0 , 0 ) denoting the number of out- and in-neighbours of the centre vertex. For p , q , d ∈ N ∪ { ∞ } , we denote by S p , q , d the family of stars ( k , ℓ ) such that k ⩽ p , and ℓ ⩽ q , and 0 < k + ℓ ⩽ d . We prove the S - Packing problem is polynomial if S = S p , q , d for some p , q , d ∈ Z + ∪ { ∞ } , and NP-complete otherwise.