Union of a spanning arborescence and a directed spanning tree

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Find a good characterization of directed graphs having two disjoint directed spanning trees such that one of the spanning trees is an arborescence rooted at a given node.

Solved b.jpg

It was shown by Jørgen Bang-Jensen and Anders Yeo [1] that the problem is NP-complete. They also showed that it is NP-complete to decide if a strongly connected digraph contains a spanning strong subdigraph so that removing these arcs the rest stays connected.


This problem was proposed by Stéphan Thomassé. As a generalization, one may ask for k disjoint directed spanning trees, of which l are arborescences rooted at a given node.

Note that there are well-known characterizations of graphs that have k edge-disjoint spanning trees (Tutte's disjoint tree theorem [2]), as well as of digraphs that have k edge-disjoint spanning arborescences rooted at a given node (Edmonds' disjoint arborescences theorem [3]).

Related questions

Kriesell and Bang-Jensen [4] proved that for a strongly connected digraph D it can be decided in polynomial time whether the underlying undirected graph contains two node-disjoint cycles, one of which is a directed cycle in D. However, the same problem is NP-complete if we drop the requirement of strong connectivity [5].

Another related problem is the existence of Disjoint spanning in- and out-arborescences.


  1. J. Bang-Jensen, A. Yeo, Arc-disjoint spanning sub(di)graphs in Digraphs, DOI link, author link.
  2. W.T. Tutte, On the problem of decomposing a graph into n connected factors, J. London Math. Soc. 36 (1961) 221-230. DOI link
  3. J. Edmonds, Edge-disjoint branchings, in: B. Rustin, ed., Combinatorial Algorithms (Acad. Press, New York, 1973) 91-96.
  4. M. Kriesell, J. Bang-Jensen, On the problem of finding disjoint cycles and dicycles in a digraph, IMADA Preprint PP-2009-12
  5. J. Bang-Jensen, M. Kriesell, A. Maddaloni, S. Simonsen, Vertex-disjoint directed and undirected cycles in general digraphs, arXiv link.