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Finding good 2-partitions of digraphs. II. Enumerable properties. (English) Zbl 1345.68168
Summary: We continue the study, initiated in Part I [the first and the last author, ibid. 636, 85–94 (2016; Zbl 1342.68150)], 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 $$\mathcal{E}$$ be the following set of properties of digraphs: $$\mathcal{E} =\{$$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 $$\mathbb{P}_1$$, $$\mathbb{P}_2$$ from $$\mathcal{E}$$ and all pairs of fixed positive integers $$k_1$$, $$k_2$$, the complexity of deciding whether a digraph has a vertex partition into two digraphs $$D_1$$, $$D_2$$ such that $$D_i$$ has property $$\mathbb{P}_i$$ and $$| V(D_i) | \geq k_i$$, $$i = 1, 2$$. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the analogous problems when $$\mathbb{P}_1 \in \mathcal{H}$$ and $$\mathbb{P}_2 \in \mathcal{H} \cup \mathcal{E}$$, where $$\mathcal{H} =\{\text{acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament}\}$$ were completely characterized in [loc. cit.].

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 05C20 Directed graphs (digraphs), tournaments 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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