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Algorithms for finding an induced cycle in planar graphs

Content Provider	Springer Nature Link
Author	Kawarabayashi, Ken ichi Kobayashi, Yusuke
Copyright Year	2010
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.Our main results in this paper are to improve their result in the following sense. 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.
Ending Page	734
Page Count	20
Starting Page	715
File Format	PDF
ISSN	02099683
e-ISSN	14396912
Journal	Combinatorica
Issue Number	6
Volume Number	30
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-03-19
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematics Graph algorithms Combinatorics Paths and cycles Graph minors
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Computational Mathematics

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