On a Disparity Between Relative Cliquewidth and Relative NLC - width by Haiko Müller & Ruth Urner December 2008

Cliquewidth and NLC-width are two closely related parameters that measure the complexity of graphs. Both cliqueand NLC-width are defined to be the minimum number of labels required to create a labelled graph by certain terms of operations. Many hard problems on graphs become solvable in polynomial time if the inputs are restricted to graphs of bounded cliqueor NLC-width. Cliquewidth and NLC-width differ at most by a factor of two. The relative counterparts of these parameters are defined to be the minimum number of labels necessary to create a graph while the tree-structure of the term is fixed. We show that Relative Cliquewidth and Relative NLC-width differ significantly in computational complexity. While the former problem is NP-complete the latter is solvable in polynomial time. The relative NLC-width can be computed in Opn3q time, which also yields an exact algorithm for computing the NLC-width in time Op3nnq. Additionally, our technique enables a combinatorial characterisation of NLC-width that avoids the usual operations on labelled graphs.