Finding good 2-partitions of digraphs II. Enumerable properties

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 P 1 , P 2 from 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 P i and | V ( D i ) | ź 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 P 1 ź H and P 2 ź H ź E , where H = {acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in 3.

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