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 longstanding 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 NPcomplete 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 nontrivially 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 NPcomplete, and so we need to impose some condition on k.
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Kawarabayashi, Ki., Kobayashi, Y. Algorithms for finding an induced cycle in planar graphs. Combinatorica 30, 715–734 (2010). https://doi.org/10.1007/s004930102499x
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Mathematics Subject Classification (2000)
 05C85
 05C38
 05C83