# Algorithms for finding an induced cycle in planar graphs

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## Abstract

In this paper, we consider the problem of finding an induced cycle passing through *k* given vertices, which we call the *induced cycle problem*. The significance of finding induced cycles stems from the fact that a precise characterization of perfect graphs would require understanding the structure of graphs without an odd induced cycle and its complement. There has been huge progress in the recent years, especially, the Strong Perfect Graph Conjecture was solved in [6]. Concerning recognition of perfect graphs, there had been a long-standing open problem for detecting an odd hole and its complement, and finally this was solved in [4].

Unfortunately, the problem of finding an induced cycle passing through two given vertices is NP-complete in a general graph [2]. However, if the input graph is constrained to be planar and *k* is fixed, then the induced cycle problem can be solved in polynomial time [11]–[13]. In particular, an O(*n* ^{2}) time algorithm is given for the case *k*=2 by McDiarmid, Reed, Schrijver and Shepherd [14], where *n* is the number of vertices of the input graph.

- 1.
The number of vertices

*k*is allowed to be non-trivially super constant number, up to \( k = o\left( {\left( {\tfrac{{\log n}} {{\log \log n}}} \right)^{\tfrac{2} {3}} } \right) \). More precisely, when \( k = o\left( {\left( {\tfrac{{\log n}} {{\log \log n}}} \right)^{\tfrac{2} {3}} } \right) \), then the ICP can be solved in O(*n*^{2+ɛ }) time for any*ɛ*> 0. - 2.
The time complexity is linear if

*k*is fixed.

We note that the linear time algorithm (the second result) is independent from the first result.

Let us observe that if *k* is as a part of the input, then the problem is still NP-complete, and so we need to impose some condition on *k*.

## Mathematics Subject Classification (2000)

05C85 05C38 05C83## Preview

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## References

- [1]S. Arnborg and A. Proskurowski: Linear time algorithms for NP-hard problems restricted to partial
*k*-trees,*Discrete Applied Math*.**23**(1989), 11–24.CrossRefzbMATHMathSciNetGoogle Scholar - [2]D. Bienstock: On the complexity of testing for even holes and induced odd paths,
*Discrete Math*.**90**(1991), 85–92.CrossRefzbMATHMathSciNetGoogle Scholar - [3]H. L. Bodlaender: A linear time algorithm for finding tree-decompositions of small treewidth,
*SIAM Journal on Computing***25**(1996), 1305–1317.CrossRefzbMATHMathSciNetGoogle Scholar - [4]M. Chudnovsky, G. Cornuéjols, X. Liu, P. D. Seymour and K. Vušković: Recognizing Berge graphs,
*Combinatorica***25(2)**(2005), 143–186.CrossRefzbMATHMathSciNetGoogle Scholar - [5]M. Chudnovsky, K. Kawarabayashi and P. D. Seymour: Detecting even holes,
*Journal of Graph Theory***48**(2005), 85–111.CrossRefzbMATHMathSciNetGoogle Scholar - [6]M. Chudnovsky, N. Robertson, P. D. Seymour and R. Thomas: The strong perfect graph theorem,
*Annals of Mathematics***64**(2006), 51–219.CrossRefMathSciNetGoogle Scholar - [7]M. Conforti, G. Cornuéjols, A. Kapoor and K. Vušković: Even-hole-free graphs. I. Decomposition theorem;
*Journal of Graph Theory***39**(2002), 6–49.CrossRefzbMATHMathSciNetGoogle Scholar - [8]M. Conforti, G. Cornuéjols, A. Kapoor and K. Vušković: Even-hole-free graphs, II. Recognition algorithm;
*Journal of Graph Theory***40**(2002), 238–266.CrossRefzbMATHMathSciNetGoogle Scholar - [9]M. R. Fellows: The Robertson-Seymour Theorems: a survey of applications; in:
*Comtemporary Mathematics*, Vol.**89**, pp. 1–18, American Mathematical Society, 1987.MathSciNetGoogle Scholar - [10]M. R. Fellows, J. Kratochvil, M. Middendorf and F. Pfeiffer: The complexity of induced minors and related problems,
*Algorithmica***13**(1995), 266–282.CrossRefzbMATHMathSciNetGoogle Scholar - [11]K. Kawarabayashi and Y. Kobayashi: A linear time algorithm for the induced disjoint paths problem in planar graphs, manuscript.Google Scholar
- [12]K. Kawarabayashi and Y. Kobayashi: The induced disjoint paths problem, in:
*Proceedings of the 13th Integer Programming and Combinatorial Optimization Conference (LNCS 5035)*, 2008, pp. 47–61.Google Scholar - [13]Y. Kobayashi: Induced disjoint paths problem in a planar digraph,
*Discrete Applied Math*.**157**(2009), pp. 3231–3238.CrossRefzbMATHGoogle Scholar - [14]C. McDiarmid, B. Reed, A. Schrijver and B. Shepherd: Induced circuits in planar graphs,
*Journal of Combinatorial Theory Ser. B***60**(1994), 169–176.CrossRefzbMATHMathSciNetGoogle Scholar - [15]B. Mohar and C. Thomassen:
*Graphs on Surfaces*, The Johns Hopkins University Press, Baltimore and London, 2001.zbMATHGoogle Scholar - [16]B. Reed: Rooted routing in the plane,
*Discrete Applied Math*.**57**(1995), 213–227.CrossRefzbMATHGoogle Scholar - [17]B. A. Reed, N. Robertson, A. Schrijver and P. D. Seymour: Finding disjoint trees in planar graphs in linear time, in:
*Contemporary Mathematics*, Vol.**147**, pp. 295–301, American Mathematical Society, 1993.MathSciNetGoogle Scholar - [18]N. Robertson and P. D. Seymour: Graph minors, VII. Disjoint paths on a surface;
*Journal of Combinatorial Theory Ser. B***45**(1988), 212–254.CrossRefzbMATHMathSciNetGoogle Scholar - [19]N. Robertson and P. D. Seymour: Graph minors, XI. Circuits on a surface;
*Journal of Combinatorial Theory Ser. B***60**(1994), 72–106.CrossRefzbMATHMathSciNetGoogle Scholar - [20]N. Robertson and P. D. Seymour: Graph minors, XIII. The disjoint paths problem;
*Journal of Combinatorial Theory Ser. B***63**(1995), 65–110.CrossRefzbMATHMathSciNetGoogle Scholar - [21]N. Robertson and P. D. Seymour: Graph minors. XXII. Irrelevant vertices in linkage problems, manuscript.Google Scholar
- [22]N. Robertson, P. D. Seymour and R. Thomas: Quickly excluding a planar graph,
*Journal of Combinatorial Theory Ser. B***62**(1994), 323–348.CrossRefzbMATHMathSciNetGoogle Scholar