In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on Pt\documentclass[12pt]{minimal}
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\begin{document}$$P_t$$\end{document}-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t≤5\documentclass[12pt]{minimal}
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\begin{document}$$t\le 5$$\end{document} (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 t=6\documentclass[12pt]{minimal}
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\begin{document}$$t=6$$\end{document} announced recently (Grzesik et al. in Polynomial-time algorithm for maximum weight independent set on P6\documentclass[12pt]{minimal}
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\begin{document}$${P}_6$$\end{document}-free graphs. CoRR, arXiv:1707.05491, 2017). Here we study the existence of subexponential-time algorithms for the problem: we show that for any t≥1\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 1$$\end{document}, there is an algorithm for Maximum Independent Set on Pt\documentclass[12pt]{minimal}
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\begin{document}$$P_t$$\end{document}-free graphs whose running time is subexponential in the number of vertices. Even for the weighted version MWIS, the problem is solvable in 2O(tnlogn)\documentclass[12pt]{minimal}
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\begin{document}$$2^{\mathcal {O}(\sqrt{tn \log n})}$$\end{document} time on Pt\documentclass[12pt]{minimal}
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\begin{document}$$P_t$$\end{document}-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 2O(|V(H)|n+mlog(n+m))\documentclass[12pt]{minimal}
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\begin{document}$$2^{\mathcal {O}(|V(H)|\sqrt{n+m}\log (n+m))}$$\end{document}, even if d is part of the input.Otherwise, assuming the Exponential-Time Hypothesis (ETH), there is no 2o(n+m)\documentclass[12pt]{minimal}
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\begin{document}$$2^{o(n+m)}$$\end{document}-time algorithm for d-Scattered Set for any fixed d≥3\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 3$$\end{document} on H-free graphs with n-vertices and m-edges.
