Bipartite spanning sub(di)graphs induced by 2-partitions

For a given $2$-partition $(V1,V2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $BG(V1,V2)$ of $G$ induced by those arcs/edges that have one end in each $Vi$. We determine, for all pairs of non-negative integers $k1,k2$, the complexity of deciding whether $G$ has a 2-partition $(V1,V2)$ such that each vertex in $Vi$ has at least $ki$ (out-)neighbours in $V{3-i}$. We prove that it is ${\cal NP}$-complete to decide whether a digraph $D$ has a 2-partition $(V1,V2)$ such that each vertex in $V1$ has an out-neighbour in $V2$ and each vertex in $V2$ has an in-neighbour in $V1$. The problem becomes polynomially solvable if we require $D$ to be strongly connected. We give a characterisation, based on the so-called strong component digraph of a non-strong digraph of the structure of ${\cal NP}$-complete instances in terms of their strong component digraph. When we want higher in-degree or out-degree to/from the other set the problem becomes ${\cal NP}$-complete even for strong digraphs. A further result is that it is ${\cal NP}$-complete to decide whether a given digraph $D$ has a $2$-partition $(V1,V2)$ such that $BD(V1,V2)$ is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.