Finding good 2-partitions of digraphs I. Hereditary properties.

We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote the following two sets of natural properties of digraphs: H = {acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and 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 the complexity of deciding, for any fixed pair of positive integers k 1 , k 2 , whether a given digraph has a vertex partition into two digraphs D 1 , D 2 such that | V ( D i ) | ź k i and D i has property P i for i = 1 , 2 when P 1 ź H and P 2 ź H ź E . We also classify the complexity of the same problems when restricted to strongly connected digraphs.The complexity of the 2-partition problems where both P 1 and P 2 are in E is determined in the companion paper 2.