On a disparity between relative cliquewidth and relative NLC-width

Authors:
Haiko Müller;Ruth Urner
Affiliations:
School of Computing, University of Leeds, Leeds, LS2 9JT, United Kingdom;David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Venue:
Discrete Applied Mathematics
Year:
2010

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Abstract

Cliquewidth and NLC-width are two closely related parameters that measure the complexity of graphs. Both clique- and 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 clique- or 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 O(n^3) time, which also yields an exact algorithm for computing the NLC-width in time O(3^nn). Additionally, our technique enables a combinatorial characterisation of NLC-width that avoids the usual operations on labelled graphs.