Abstract
We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete even for bipartite series-parallel graphs, which form a proper subclass of bipartite planar graphs. We note that our reduction indeed shows the PSPACE-completeness for graphs with pathwidth two, and it can be extended for threshold graphs. In contrast, we give a polynomial-time algorithm to solve the problem for graphs with pathwidth one. Thus, this paper gives precise analyses of the problem with respect to pathwidth.
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
References
Bonamy, M., Johnson, M., Lignos, I., Patel, V., Paulusma, D.: Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim. 27, 132–143 (2014)
Bonamy, M., Bousquet, N.: Recoloring bounded treewidth graphs. Electron. Notes Discrete Math. 44, 257–262 (2013)
Bonsma, P.: Rerouting shortest paths in planar graphs. In: Proceedings of FSTTCS 2012, LIPIcs 18, pp. 337–349 (2012)
Bonsma, P.: The complexity of rerouting shortest paths. Theoret. Comput. Sci. 510, 1–12 (2013)
Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoret. Comput. Sci. 410, 5215–5226 (2009)
Bonsma, P., Kamiński, M., Wrochna, M.: Reconfiguring independent sets in claw-free graphs. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 86–97. Springer, Heidelberg (2014)
Bonsma, P., Mouawad, A.E.: The complexity of bounded length graph recoloring (2014). arXiv:1404.0337
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)
Cereceda, L.: Mixing graph colourings. Ph.D. thesis, London School of Economics and Political Science (2007)
Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between \(3\)-colorings. J. Graph Theor. 67, 69–82 (2011)
Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of Boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38, 2330–2355 (2009)
Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoret. Comput. Sci. 343, 72–96 (2005)
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theoret. Comput. Sci. 412, 1054–1065 (2011)
Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. Discrete Appl. Math. 160, 2199–2207 (2012)
Ito, T., Kawamura, K., Ono, H., Zhou, X.: Reconfiguration of list \(L(2,1)\)-labelings in a graph. Theoret. Comput. Sci. 544, 84–97 (2014)
Ito, T., Kawamura, K., Zhou, X.: An improved sufficient condition for reconfiguration of list edge-colorings in a tree. IEICE Trans. Inf. Syst. E95–D, 737–745 (2012)
Jansen, K., Scheffler, P.: Generalized coloring for tree-like graphs. Discrete Appl. Math. 75, 135–155 (1997)
Johnson, M., Kratsch, D., Kratsch, S., Patel, V., Paulusma, D.: Colouring reconfiguration is fixed-parameter tractable (2014). arXiv:1403.6347
Kamiński, M., Medvedev, P., Milanič, M.: Shortest paths between shortest paths. Theoret. Comput. Sci. 412, 5205–5210 (2011)
Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theoret. Comput. Sci. 439, 9–15 (2012)
Makino, K., Tamaki, S., Yamamoto, M.: An exact algorithm for the Boolean connectivity problem for \(k\)-CNF. Theoret. Comput. Sci. 412, 4613–4618 (2011)
Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 281–294. Springer, Heidelberg (2013)
Proskurowski, A., Telle, J.A.: Classes of graphs with restricted interval models. Discrete Math. Theor. Comput. Sci. 3, 167–176 (1999)
Robertson, N., Seymour, P.: Graph minors. I Excluding a forest. J. Comb. Theor. Ser. B 35, 39–61 (1983)
Wrochna, M.: Reconfiguration in bounded bandwidth and treedepth (2014). arXiv:1405.0847
Acknowledgments
We are grateful to Daichi Fukase and Yuma Tamura for fruitful discussions with them. This work is partially supported by JSPS KAKENHI Grant Numbers 25106504 and 25330003.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Hatanaka, T., Ito, T., Zhou, X. (2014). The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-12691-3_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12690-6
Online ISBN: 978-3-319-12691-3
eBook Packages: Computer ScienceComputer Science (R0)