Extended skew partition problem - Dantas, Simone; de Figueiredo, Celina M.H.; Gravier, Sylvain; Klein, Sulamita (Discrete mathematics, 2006)
A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts A 1 , A 2 , B 1 , B 2 such that there are all possible edges between A 1 and A 2 , and no edges between B 1 and B 2 . We introduce the concept of ( n 1 , n 2 ) -extended skew partition which includes all partitioning problems into n 1 + n 2 nonempty parts A 1 , … , A n 1 , B 1 , … , B n 2 such that there are all possible edges between the A i parts, no edges between the B j parts, i ∈ { 1 , … , n 1 } , j ∈ { 1 , … , n 2 } , which generalizes the skew partition. We present a polynomial-time algorithm for testing whether a graph admits an ( n 1 , n 2 ) -extended skew partition. As a tool to complete this task we also develop a generalized 2-SAT algorithm, which by itself may have application to other partition problems.