# Chordless Paths Through Three Vertices

• Robert Haas
• Michael Hoffmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3162)

## Abstract

Consider the following problem, that we call “Chordless Path through Three Vertices” or Cp3v, for short: Given a simple undirected graph G=(V,E), a positive integer k, and three distinct vertices s, t, and vV, is there a chordless path from s via v to t in G that consists of at most k vertices? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternatively, one could say that the subgraph induced by the vertex set of the path in G is the path itself. The problem has been raised in the context of service deployment in communication networks. We resolve the parametric complexity of Cp3v by proving it W[1]-complete with respect to its natural parameter k. Our reduction extends to a number of related problems about chordless paths. In particular, deciding on the existence of a single directed chordless (s,t)-path in a digraph is also W[1]-complete with respect to the length of the path.

## Keywords

graph theory induced path parameterized complexity

## References

1. 1.
Bazgan, C.: Schémas d’approximation et complexité paramétrée. Rapport du stage (DEA), Université Paris Sud (1995)Google Scholar
2. 2.
Berge, C.: Färbung von Graphen deren sämtliche beziehungsweise deren ungerade Kreise starr sind (Zusammenfassung). Wiss. Z.Martin Luther Univ. HalleWittenberg Math. Naturwiss. Reihe, 114–115 (1961)Google Scholar
3. 3.
Bienstock, D.: On the complexity of testing for odd holes and induces odd paths. Discrete Math. 90(1), 85–92 (1991)Google Scholar
4. 4.
Bienstock, D.: Corrigendum to: On the complexity of testing for odd holes and induces odd paths. Discrete Math. 102(1), 109 (1992)Google Scholar
5. 5.
Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: On the parameterized complexity of short computation and factorization. Arch. Math. Logic 36(4–5), 321–337 (1997)
6. 6.
Cesati, M.: Perfect code is W[1]-complete. Inform. Process. Lett. 81(3), 163–168 (2002)
7. 7.
Cesati, M.: The Turing way to parameterized complexity. J. Comput. Syst. Sci. 67(4), 654–685 (2003)
8. 8.
Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Inform. Process. Lett. 64(4), 165–171 (1997)
9. 9.
Chudnovsky, M., Robertson, N., Seymour, P.D., Thomas, R.: The strong perfect graph theorem (2003) (manuscript)Google Scholar
10. 10.
Chudnovsky, M., Seymour, P.D.: Recognizing Berge graphs (2003) (manuscript)Google Scholar
11. 11.
CornuéJols, G., Liu, X., Vušković, K.: A polynomial algorithm for recognizing perfect graphs. In: Proc. 44th Annu. IEEE Sympos. Found. Comput. Sci., pp. 20–27 (2003)Google Scholar
12. 12.
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoret. Comput. Sci. 141, 109–131 (1995)
13. 13.
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Heidelberg (1999)Google Scholar
14. 14.
Downey, R.G., Fellows, M.R., Kapron, B., Hallett, M.T., Wareham, H.T.: Parameterized complexity and some problems in logic and linguistics. In: Matiyasevich, Y.V., Nerode, A. (eds.) LFCS 1994. LNCS, vol. 813, pp. 89–101. Springer, Heidelberg (1994)Google Scholar
15. 15.
Fellows, M.R.: The Robertson-Seymour theorems: A survey of applications. In: Proc. AMS-IMS-SIAM Joint Summer Research Conf., Providence, RI, pp. 1–18 (1989)Google Scholar
16. 16.
Fellows, M.R., Kratochvíl, J., Middendorf, M., Pfeiffer, F.: The complexity of induced minors and related problems. Algorithmica 13, 266–282 (1995)
17. 17.
Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Canad. J. Math. 8, 399–404 (1956)
18. 18.
Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121 (1980)
19. 19.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
20. 20.
Haas, R.: Service Deployment in Programmable Networks. PhD thesis, ETH Zurich, Switzerland (2003)Google Scholar
21. 21.
Karp, R.M.: On the complexity of combinatorial problems. Networks 5, 45–68 (1975)
22. 22.
Lueker, G.S., Rose, D.J., Tarjan, R.E.: Algorithmic aspects of vertex elimination in graphs. SIAM J. Comput. 5, 266–283 (1976)
23. 23.
Mcdiarmid, C., Reed, B., Schrijver, A., Shepherd, B.: Non-interfering network flows. In: Proc. 3rd Scand. Workshop Algorithm Theory, pp. 245–257 (1992)Google Scholar
24. 24.
Mcdiarmid, C., Reed, B., Schrijver, A., Shepherd, B.: Induced circuits in planar graphs. J. Combin. Theory Ser. B 60, 169–176 (1994)
25. 25.
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Habilitation thesis, Wilhelm-Schickard Institut für Informatik, Universität Tübingen, Germany (2002)Google Scholar
26. 26.
Nikolopoulos, S.D., Palios, L.: Hole and antihole detection in graphs. In: Proc. 15th ACM-SIAM Sympos. Discrete Algorithms, pp. 843–852 (2004)Google Scholar
27. 27.
Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63, 65–110 (1995)
28. 28.
Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13, 566–579 (1984)