## Abstract

Let us say a graph is O_{s}-free, where s≥1 is an integer, if there do not exist s cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when s=2, is not well understood. For instance, until now we did not know how to test whether a graph is O_{2}-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that O_{2}-free graphs have only a polynomial number of induced paths. In this paper we prove Le's conjecture; indeed, we will show that for all s≥1, there exists c>0 such that every O_{s}-free graph G has at most |G|^{c} induced paths, where |G| is the number of vertices. This provides a poly-time algorithm to test if a graph is O_{s}-free, for all fixed s. The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every O_{s}-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in |G|. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.

Original language | English (US) |
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Pages (from-to) | 321-339 |

Number of pages | 19 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 164 |

DOIs | |

State | Published - Jan 2024 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Induced subgraphs