Abstract
AbstractLet be a digraph with a designated root vertex . Edmonds’ seminal result (see J. Edmonds [4]) implies that has a packing of spanning ‐arborescences if and only if has a packing of ‐paths for all , where a packing means arc‐disjoint subgraphs. Let be a matroid on the set of arcs leaving . A packing of ‐paths is called ‐based if their arcs leaving form a base of while a packing of ‐arborescences is called ‐based if, for all , the packing of ‐paths provided by the arborescences is ‐based. Durand de Gevigney, Nguyen, and Szigeti proved in [3] that has an ‐based packing of ‐arborescences if and only if has an ‐based packing of ‐paths for all . Bérczi and Frank conjectured that this statement can be strengthened in the sense of Edmonds’ theorem such that each ‐arborescence is required to be spanning. Specifically, they conjectured that has an ‐based packing of spanning ‐arborescences if and only if has an ‐based packing of ‐paths for all . In this paper we disprove this conjecture in its general form and we prove that the corresponding decision problem is NP‐complete. We also prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids. For all the results presented in this paper, the undirected counterpart also holds.
Highlights
The packing problem in digraphs is one of the fundamental topics in graph theory and combinatorial optimization, where the goal is to find the largest family of disjoint subgraphs satisfying a specified property in a given digraph
There exists a packing of k spanning s‐arborescences in a rooted digraph D = (V + s, A) if and only if there exists a packing of k (s, t)‐paths in D for every t ∈ V
We will give a counterexample to Conjecture 1.3 and prove that Problem 1.4 is NP‐complete for acyclic digraphs and a certain class of matroids
Summary
The packing problem in digraphs is one of the fundamental topics in graph theory and combinatorial optimization, where the goal is to find the largest family of disjoint subgraphs satisfying a specified property in a given digraph. If D has a packing of k spanning s‐arborescences, D has a packing of k (s, t)‐paths for every t ∈ V , since each of the arborescences contains an (s, t)‐path. There exists a packing of k spanning s‐arborescences in a rooted digraph D = (V + s, A) if and only if there exists a packing of k (s, t)‐paths in D for every t ∈ V. We are interested in a packing of (s, t)‐paths whose root arcs form a base of k. There exists an k‐based packing of spanning s‐arborescences in D if and only if there exists an k‐based packing of (s, t)‐paths in D for every t ∈ V. Given a matroid‐rooted digraph (D = (V + s, A), k), decide whether there exists an k‐based packing of spanning s‐arborescences in D. We will prove that Conjecture 1.3 is true for several fundamental classes of matroids such as graphic and transversal matroids
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