Abstract
A modulator of a graph G to a specified graph class \({\mathcal H}\) is a set of vertices whose deletion puts G into \({\mathcal H}\). The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but “well-structured” (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently?
We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for efficiently deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.
Supported by the Austrian Science Fund (FWF), project P26696.
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
Alekseev, V.E.: Polynomial algorithm for finding the largest independent sets in graphs without forks. Discr. Appl. Math. 135(1–3), 3–16 (2004)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernel bounds for path and cycle problems. Theor. Comput. Sci. 511, 117–136 (2013)
Brandstädt, A., Lozin, V.V.: A note on alpha-redundant vertices in graphs. Discr. Appl. Math. 108(3), 301–308 (2001)
Cai, L.: Parameterized complexity of vertex colouring. Discr. Appl. Math. 127(3), 415–429 (2003)
Corneil, D.G., Lerchs, H., Burlingham, L.S.: Complement reducible graphs. Discr. Appl. Math. 3, 163–174 (1981)
Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)
Cunningham, W.H.: Decomposition of directed graphs. SIAM J. Algebraic Discrete Methods 3(2), 214–228 (1982)
Diestel, R.: Graph Theory. GTM, vol. 173, 2nd edn. Springer Verlag, New York (2000)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer Verlag (2013)
Gajarský, J., Hliněný, P., Obdržálek, J., Ordyniak, S., Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S.: Kernelization using structural parameters on sparse graph classes. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 529–540. Springer, Heidelberg (2013)
Ganian, R., Hliněný, P.: On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width. Discr. Appl. Math. 158(7), 851–867 (2010)
Gaspers, S., Misra, N., Ordyniak, S., Szeider, S., Živný, S.: Backdoors into heterogeneous classes of SAT and CSP. In: Brodley, C.E., Stone, P.(eds.), Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 2652–2658. AAAI Press (2014)
Gerber, M.U., Lozin, V.V.: Robust algorithms for the stable set problem. Graphs and Combinatorics 19(3), 347–356 (2003)
Gioan, E., Paul, C.: Dynamic distance hereditary graphs using split decomposition. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 41–51. Springer, Heidelberg (2007)
Gioan, E., Paul, C.: Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs. Discr. Appl. Math. 160(6), 708–733 (2012)
Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient split decomposition via graph-labelled trees. Algorithmica 69(4), 789–843 (2014)
Golovach, P.A., Paulusma, D., Song, J.: Closing complexity gaps for coloring problems on h-free graphs. Inf. Comput. 237, 204–214 (2014)
Hliněný, P., Oum, S.I.: Finding branch-decompositions and rank-decompositions. SIAM J. Comput. 38(3), 1012–1032 (2008)
Kochol, M., Lozin, V.V., Randerath, B.: The 3-colorability problem on graphs with maximum degree four. SIAM J. Comput. 32(5), 1128–1139 (2003)
Libkin, L.: Elements of Finite Model Theory. Springer (2004)
Lokshantov, D., Vatshelle, M., Villanger, Y.: Independent set in p\(_{\text{5 }}\)-free graphs in polynomial time. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pp. 570–581. SIAM (2014)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006)
Oum, S., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theory Ser. B 96(4), 514–528 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Eiben, E., Ganian, R., Szeider, S. (2015). Solving Problems on Graphs of High Rank-Width. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-21840-3_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21839-7
Online ISBN: 978-3-319-21840-3
eBook Packages: Computer ScienceComputer Science (R0)