### Abstract

In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on (Formula presented.)-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for (Formula presented.) (Lokshtanov et al., in: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms, SODA 2014, Portland, OR, USA, January 5–7, 2014, pp 570–581, 2014), and an algorithm for (Formula presented.) announced recently (Grzesik et al. in Polynomial-time algorithm for maximum weight independent set on (Formula presented.)-free graphs. CoRR, arXiv:1707.05491, 2017). Here we study the existence of subexponential-time algorithms for the problem: we show that for any (Formula presented.), there is an algorithm for Maximum Independent Set on (Formula presented.)-free graphs whose running time is subexponential in the number of vertices. Even for the weighted version MWIS, the problem is solvable in (Formula presented.) time on (Formula presented.)-free graphs. For approximation of MIS in broom-free graphs, a similar time bound is proved. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least d from each other. We give a complete characterization of those graphs H for which d-Scattered Set on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges):If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time (Formula presented.), even if d is part of the input.Otherwise, assuming the Exponential-Time Hypothesis (ETH), there is no (Formula presented.)-time algorithm for d-Scattered Set for any fixed (Formula presented.) on H-free graphs with n-vertices and m-edges.

Original language | English |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Algorithmica |

DOIs | |

Publication status | Accepted/In press - Jul 16 2018 |

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### Keywords

- Approximation
- H-free graphs
- Independent set
- Scattered set
- Subexponential algorithms

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica*, 1-18. https://doi.org/10.1007/s00453-018-0479-5