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

Abstract

AbstractFor a given ‐partition of the vertices of a (di)graph , we study properties of the spanning bipartite subdigraph of induced by those arcs/edges that have one end in each . We determine, for all pairs of nonnegative integers , the complexity of deciding whether has a 2‐partition such that each vertex in (for ) has at least (out‐)neighbours in . We prove that it is ‐complete to decide whether a digraph has a 2‐partition such that each vertex in has an out‐neighbour in and each vertex in has an in‐neighbour in . The problem becomes polynomially solvable if we require to be strongly connected. We give a characterisation of the structure of ‐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 ‐complete even for strong digraphs. A further result is that it is ‐complete to decide whether a given digraph has a ‐partition such that is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.