eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-11-28
49:1
49:22
10.4230/LIPIcs.ISAAC.2019.49
article
When Maximum Stable Set Can Be Solved in FPT Time
Bonnet, Édouard
1
https://orcid.org/0000-0002-1653-5822
Bousquet, Nicolas
2
Thomassé, Stéphan
1
3
Watrigant, Rémi
1
https://orcid.org/0000-0002-6243-5910
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
CNRS, G-SCOP laboratory, Grenoble-INP, France
Institut Universitaire de France
Maximum Independent Set (MIS for short) is in general graphs the paradigmatic W[1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in H-free graphs. We show that for every fixed t >=slant 1, MIS is FPT in P(1,t,t,t)-free graphs, where P(1,t,t,t) is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. We also provide randomized FPT algorithms in dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard dichotomy for five-vertex graphs H.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol149-isaac2019/LIPIcs.ISAAC.2019.49/LIPIcs.ISAAC.2019.49.pdf
Parameterized Algorithms
Independent Set
H-Free Graphs