Advertisement

Tropical Paths in Vertex-Colored Graphs

  • Johanne Cohen
  • Giuseppe F. Italiano
  • Yannis Manoussakis
  • Kim Thang Nguyen
  • Hong Phong PhamEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10628)

Abstract

A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the initial graph. In this work we study the problem of finding tropical paths in vertex-colored graphs. There are two versions for this problem: the shortest tropical path problem (STPP), i.e., finding a tropical path with the minimum total weight, and the maximum tropical path problem (MTPP), i.e., finding a path with the maximum number of colors possible. We show that both versions of this problems are NP-hard for directed acyclic graphs, cactus graphs and interval graphs. Moreover, we also provide a fixed parameter algorithm for STPP in general graphs and several polynomial-time algorithms for MTPP in specific graphs, including bipartite chain graphs, threshold graphs, trees, block graphs, and proper interval graphs.

References

  1. 1.
    Akbari, S., Liaghat, V., Nikzad, A.: Colorful paths in vertex coloring of graphs. Electron. J. Comb. 18(1), P17 (2011)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Anglés d’Auriac, J.-A., Bujtás, C., El Maftouhi, H., Karpinski, M., Manoussakis, Y., Montero, L., Narayanan, N., Rosaz, L., Thapper, J., Tuza, Z.: Tropical dominating sets in vertex-coloured graphs. In: Kaykobad, M., Petreschi, R. (eds.) WALCOM 2016. LNCS, vol. 9627, pp. 17–27. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-30139-6_2 CrossRefGoogle Scholar
  3. 3.
    Bertossi, A.A.: Finding hamiltonian circuits in proper interval graphs. Inform. Process. Lett. 17(2), 97–101 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bruckner, S., Hüffner, F., Komusiewicz, C., Niedermeier, R.: Evaluation of ILP-based approaches for partitioning into colorful components. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 176–187. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38527-8_17 CrossRefGoogle Scholar
  5. 5.
    Cohen, J., Manoussakis, Y., Pham, H., Tuza, Z.: Tropical matchings in vertex-colored graphs. In: Latin and American Algorithms, Graphs and Optimization Symposium (2017)Google Scholar
  6. 6.
    Corel, E., Pitschi, F., Morgenstern, B.: A min-cut algorithm for the consistency problem in multiple sequence alignment. Bioinformatics 26(8), 1015–1021 (2010)CrossRefGoogle Scholar
  7. 7.
    Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci. 77(4), 799–811 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Foucaud, F., Harutyunyan, A., Hell, P., Legay, S., Manoussakis, Y., Naserasr, R.: Tropical homomorphisms in vertex-coloured graphs. Discrete Appl. Math. 229, 1–168 (2017)Google Scholar
  9. 9.
    Ioannidou, K., Mertzios, G.B., Nikolopoulos, S.D.: The longest path problem is polynomial on interval graphs. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 403–414. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03816-7_35 CrossRefGoogle Scholar
  10. 10.
    Karger, D., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. In: Dehne, F., Sack, J.-R., Santoro, N., Whitesides, S. (eds.) WADS 1993. LNCS, vol. 709, pp. 421–432. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-57155-8_267 CrossRefGoogle Scholar
  11. 11.
    Li, H.: A generalization of the Gallai-Roy theorem. Graphs Comb. 17(4), 681–685 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lin, C.: Simple proofs of results on paths representing all colors in proper vertex-colorings. Graphs Comb. 23(2), 201–203 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Marx, D.: Graph colouring problems and their applications in scheduling. Periodica Polytech. Electr. Eng. 48(1–2), 11–16 (2004)Google Scholar
  14. 14.
    Micali, S., Vazirani, V.V.: An \({O}(\sqrt{|V|} |{E}|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of 21st Symposium on Foundations of Computer Science, pp. 17–27 (1980)Google Scholar
  15. 15.
    Uehara, R., Uno, Y.: Efficient algorithms for the longest path problem. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 871–883. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-30551-4_74 CrossRefGoogle Scholar
  16. 16.
    Uehara, R., Valiente, G.: Linear structure of bipartite permutation graphs and the longest path problem. Inform. Process. Lett. 103(2), 71–77 (2007)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Johanne Cohen
    • 1
  • Giuseppe F. Italiano
    • 2
  • Yannis Manoussakis
    • 1
  • Kim Thang Nguyen
    • 3
  • Hong Phong Pham
    • 1
    Email author
  1. 1.LRIUniversity Paris-SaclayOrsayFrance
  2. 2.Department of Civil Engineering and Computer Science EngineeringUniversity of Rome “Tor Vergata”RomeItaly
  3. 3.IBISCUniversity Paris-SaclayEvryFrance

Personalised recommendations