Finding good 2-partitions of digraphs II. Enumerable properties

• Published in: Theoretical computer science Vol. 640; pp. 1 - 19
• Main Authors: Bang-Jensen, J., Cohen, Nathann, Havet, Frédéric
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 09.08.2016; Elsevier
• Subjects: 2-Partition; Acyclic; Classification; Combinatorics; Complexity; Computational Complexity; Computer Science; Data Structures and Algorithms; Feedback vertex set; Graph theory; Integers; Mathematics; Minimum degree; NP-complete; Oriented; Out-branching; Partition; Partitions; Polynomial; Semicomplete digraph; Splitting digraphs; Symmetry; Tournament; Polynomial; Partition; 2-Partition; Minimum degree; Feedback vertex set; Acyclic; Out-branching; Oriented; Splitting digraphs; NP-complete; Semicomplete digraph; Tournament
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2016.05.034

We continue the study, initiated in [3], of the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties and given minimum cardinality. Let E be the following set of properties of digraphs: E={strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper we determine, for all choices of P1,P2 from E and all pairs of fixed positive integers k1,k2, the complexity of deciding whether a digraph has a vertex partition into two digraphs D1,D2 such that Di has property Pi and |V(Di)|≥ki, i=1,2. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the analogous problems when P1∈H and P2∈H∪E, where H={acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in [3].