Document

# Reconfiguration of Colorable Sets in Classes of Perfect Graphs

## File

LIPIcs.SWAT.2018.27.pdf
• Filesize: 0.5 MB
• 13 pages

## Cite As

Takehiro Ito and Yota Otachi. Reconfiguration of Colorable Sets in Classes of Perfect Graphs. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SWAT.2018.27

## Abstract

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• reconfiguration
• colorable set
• perfect graph

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Louigi Addario-Berry, W. Sean Kennedy, Andrew D. King, Zhentao Li, and Bruce A. Reed. Finding a maximum-weight induced k-partite subgraph of an i-triangulated graph. Discrete Applied Mathematics, 158(7):765-770, 2010. URL: http://dx.doi.org/10.1016/j.dam.2008.08.020.
2. Eyjólfur Ingi Ásgeirsson, Magnús M. Halldórsson, and Tigran Tonoyan. Universal framework for wireless scheduling problems. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 129:1-129:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.129.
3. Matthias Bentert, René van Bevern, and Rolf Niedermeier. (Wireless) Scheduling, graph classes, and c-colorable subgraphs. CoRR, abs/1712.06481, 2017. URL: http://arxiv.org/abs/1712.06481.
4. Marthe Bonamy and Nicolas Bousquet. Token sliding on chordal graphs. In Hans L. Bodlaender and Gerhard J. Woeginger, editors, Graph-Theoretic Concepts in Computer Science - 43rd International Workshop, WG 2017, Eindhoven, The Netherlands, June 21-23, 2017, Revised Selected Papers, volume 10520 of Lecture Notes in Computer Science, pages 127-139. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-68705-6_10.
5. Paul S. Bonsma, Marcin Kaminski, and Marcin Wrochna. Reconfiguring independent sets in claw-free graphs. In R. Ravi and Inge Li Gørtz, editors, Algorithm Theory - SWAT 2014 - 14th Scandinavian Symposium and Workshops, Copenhagen, Denmark, July 2-4, 2014. Proceedings, volume 8503 of Lecture Notes in Computer Science, pages 86-97. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-08404-6_8.
6. Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. Linear-time algorithm for sliding tokens on trees. Theor. Comput. Sci., 600:132-142, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.07.037.
7. Samuel Fiorini, R. Krithika, N. S. Narayanaswamy, and Venkatesh Raman. LP approaches to improved approximation for clique transversal in perfect graphs. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer Science, pages 430-442. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_36.
8. Delbert R. Fulkerson and Oliver A. Gross. Incidence matrices and interval graphs. Pac. J. Math., 15(3):835-855, 1965. URL: http://dx.doi.org/10.2140/pjm.1965.15.835.
9. M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified np-complete graph problems. Theor. Comput. Sci., 1(3):237-267, 1976. URL: http://dx.doi.org/10.1016/0304-3975(76)90059-1.
10. Paul C. Gilmore and Alan J. Hoffman. A characterization of comparability graphs and of interval graphs. Canad. J. Math., 16:539-548, 1964. URL: http://dx.doi.org/10.4153/CJM-1964-055-5.
11. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs, volume 57 of Annals of Discrete Mathematics. North Holland, second edition, 2004.
12. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1988.
13. Robert A. Hearn and Erik D. Demaine. Pspace-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci., 343(1-2):72-96, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.05.008.
14. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theor. Comput. Sci., 412(12-14):1054-1065, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.12.005.
15. Marcin Kaminski, Paul Medvedev, and Martin Milanic. Complexity of independent set reconfigurability problems. Theor. Comput. Sci., 439:9-15, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2012.03.004.
16. R. Krithika and N. S. Narayanaswamy. Parameterized algorithms for (r, l)-partization. J. Graph Algorithms Appl., 17(2):129-146, 2013. URL: http://dx.doi.org/10.7155/jgaa.00288.
17. Daniel Lokshtanov and Amer E. Mouawad. The complexity of independent set reconfiguration on bipartite graphs. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 185-195. SIAM, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.13.
18. Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Narges Simjour, and Akira Suzuki. On the parameterized complexity of reconfiguration problems. Algorithmica, 78(1):274-297, 2017. URL: http://dx.doi.org/10.1007/s00453-016-0159-2.
19. Naomi Nishimura. Introduction to reconfiguration. Preprints, 2017. 2017090055. URL: http://dx.doi.org/10.20944/preprints201709.0055.v1.
20. Jeremy P. Spinrad. Efficient Graph Representations. Fields Institute monographs. American Mathematical Society, 2003.
21. Ryuhei Uehara and Yushi Uno. On computing longest paths in small graph classes. Int. J. Found. Comput. Sci., 18(5):911-930, 2007. URL: http://dx.doi.org/10.1142/S0129054107005054.
22. Jan van den Heuvel. The complexity of change. In Simon R. Blackburn, Stefanie Gerke, and Mark Wildon, editors, Surveys in Combinatorics 2013, volume 409 of London Mathematical Society Lecture Note Series, pages 127-160. Cambridge University Press, 2013. URL: http://dx.doi.org/10.1017/CBO9781139506748.005.
X

Feedback for Dagstuhl Publishing