vertex-set partition into nonempty parts

A graph is 2 K 2 -partitionable if its vertex set can be partitioned into four nonempty parts A , B , C , D such that each vertex of A is adjacent to each vertex of B , and each vertex of C is adjacent to each vertex of D . Determining whether an arbitrary graph is 2 K 2 -partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We establish that the 2 K 2 -partition problem parameterized by minimum degree is fixed-parameter tractable. We also show that for C 4 -free graphs, circular-arc graphs, spiders, P 4 -sparse graphs, and bipartite graphs the 2 K 2 -partition problem can be solved in polynomial time.