On Reachability Mixed Arborescence Packing

TL;DR

This paper extends the classical arborescence packing theorem to mixed graphs, providing a polynomial-time algorithm for finding edge and arc-disjoint arborescences that span reachable vertices from each root.

Contribution

It offers the first polynomial-time solution to the mixed graph arborescence packing problem, generalizing previous results from directed graphs.

Findings

Developed a polynomial-time algorithm for mixed graph arborescence packing.

Extended Edmonds' theorem to mixed graphs with both directed and undirected edges.

Provided a characterization of when such packings exist in mixed graphs.

Abstract

As a generalization of the Edmonds arborescence packing theorem, Kamiyama--Katoh--Takizawa (2009) gave a good characterization of directed graphs that contain arc-disjoint arborescences spanning the set of vertices reachable from each root. Fortier--Kir\'aly--L\'eonard--Szigeti--Talon (2018) asked whether the result can be extended to mixed graphs by allowing both directed arcs and undirected edges. In this paper, we solve this question by developing a polynomial-time algorithm for finding a collection of edge and arc-disjoint arborescences spanning the set of vertices reachable from each root in a given mixed graph.