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A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By Pt, we denote a path on t vertices. In this paper, we give polynomial-time algorithms for the following problems: the maximum weight independent set problem in long-hole--free graphs and the feedback vertex set problem in P5-free graphs. Each of the above results resolves a corresponding long-standing open problem. An extended C5 is a five-vertex hole with an additional vertex adjacent to one or two consecutive vertices of the hole. Let C be the class of graphs excluding an extended C5 and holes of length at least 6 as induced subgraphs; C contains all long-hole--free graphs and all P5-free graphs. We show that, given an n-vertex graph G\in C with vertex weights and an integer k, one can, in time, n\scrO (k) find a maximum-weight induced subgraph of G of treewidth less than k. This implies both aforementioned results. To achieve this goal, we extend the framework of potential maximal cliques (PMCs) to containers. Developed by Bouchitté and Todinca [SIAM J. Comput., 31 (2001), pp. 212--232] and extended by Fomin, Todinca, and Villanger [SIAM J. Comput., 44 (2015), pp. 54--87], this framework allows us to solve a wide variety of tasks, including finding a maximum-weight induced subgraph of treewidth less than k for fixed k, in time polynomial in the size of the graph and the number of potential maximal cliques. Further developments, tailored to solve the maximum weight independent set problem within this framework (e.g., for P5-free [Lokshtanov, Vatshelle, and Villanger, SODA 2014, pp. 570--581] or P6-free graphs [Grzesik, Klimo\v sov\'a, Pilipczuk, and Pilipczuk, ACM Trans. Algorithms, 18 (2022), pp. 4:1--4:57]), enumerate only a specifically chosen subset of all PMCs of a graph. In all aforementioned works, the final step is an involved dynamic programming algorithm whose state space is based on the considered list of PMCs. Here, we modify the dynamic programming algorithm and show that it is sufficient to consider only a container for each PMC: a superset of the maximal clique that intersects the sought solution only in the vertices of the PMC. This strengthening of the framework not only allows us to obtain our main result but also leads to significant simplifications of the reasoning in previous papers. [ABSTRACT FROM AUTHOR]

There are numerous examples of the so-called "square root phenomenon" in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter k, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in O(√ k) instead of O(k) (modulo standard complexity assumptions). We consider a classic optimization problem Subset Traveling Salesman, where we are asked to visit all the terminals T by a minimum-weight closed walk. We investigate the parameterized complexity of this problem in planar graphs, where the number k = |T| of terminals is regarded as the parameter. We show that Subset TSP can be solved in time 2O(√k log k) · nO(1) even on edge-weighted directed planar graphs. This improves upon the algorithm of Klein and Marx [SODA 2014, SIAM, Philadelphia, 2014, pp. 1812-1830] with the same running time that worked only on undirected planar graphs with polynomially large integer weights. [ABSTRACT FROM AUTHOR]