Abstract
We show that a graph has tree-width at most 4k–1 if its line graph has NLC-width or clique-width at most k, and that an incidence graph has tree-width at most k if its line graph has NLC-width or clique-width at most k. In [9] it is shown that a line graph has NLC-width at most k+2 and clique-width at most 2k+2 if the root graph has tree-width k. Using these bounds we show by a reduction from tree-width minimization that NLC-width minimization is NP-complete.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM Journal of Algebraic and Discrete Methods 8(2), 227–284 (1987)
Arnborg, S., Courcelle, B., Proskurowski, A., Seese, D.: An algebraic theory of graph reduction. Journal of the ACM 40(5), 1134–1164 (1993)
Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms 18(2), 238–255 (1995)
Corneil, D.G., Habib, M., Lanlignel, J.M., Reed, B., Rotics, U.: Polynomial time recognition of clique-width at most three graphs. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 126–134. Springer, Heidelberg (2000)
Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14(4), 926–934 (1985)
Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems 33(2), 125–150 (2000)
Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Applied Mathematics 101, 77–114 (2000)
Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Brandstädt, A., Van Le, B. (eds.) WG 2001. LNCS, vol. 2204, pp. 117–128. Springer, Heidelberg (2001)
Gurski, F., Wanke, E.: Vertex disjoint paths on clique-width bounded graphs (Extended abstract). In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 119–128. Springer, Heidelberg (2004)
Gurski, F., Wanke, E.: Line graphs of bounded clique-width Manuscript (submitted) (2005), available at http://www.acs.uni-duesseldorf.de/~gurski
Gurski, F., Wanke, E.: On the relationship between NLC-width and linear NLC-width. Manuscript, accepted for Theoretical Computer Science (2005)
Johansson, Ö.: Clique-decomposition, NLC-decomposition, and modular decomposition - relationships and results for random graphs. Congressus Numerantium 132, 39–60 (1998)
Johansson, Ö.: NLC2-decomposition in polynomial time. International Journal of Foundations of Computer Science 11(3), 373–395 (2000)
Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics 126(2-3), 197–221 (2003)
Lehot, P.G.H.: An optimal algorithm to detect a line graph and output its root graph. Journal of the ACM 21(4), 569–575 (1974)
Lozin, V., Rautenbach, D.: The tree- and clique-width of bipartite graphs in special classes. Technical Report RRR 33-2004, Rutgers University (2004)
Robertson, N., Seymour, P.D.: Graph minors I. Excluding a forest. Journal of Combinatorial Theory, Series B 35, 39–61 (1983)
Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree width. Journal of Algorithms 7, 309–322 (1986)
Wanke, E.: k-NLC graphs and polynomial algorithms. Discrete Applied Mathematics 54, 251–266 (1994)
Whitney, H.: Congruent graphs and the connectivity of graphs. American Journal of Mathematics 54, 150–168 (1932)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gurski, F., Wanke, E. (2005). Minimizing NLC-Width is NP-Complete. In: Kratsch, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2005. Lecture Notes in Computer Science, vol 3787. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11604686_7
Download citation
DOI: https://doi.org/10.1007/11604686_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31000-6
Online ISBN: 978-3-540-31468-4
eBook Packages: Computer ScienceComputer Science (R0)