When Maximum Stable Set can be solved in FPT time

TL;DR

This paper introduces parameterized noise variants of co-graphs and develops FPT algorithms for Maximum Independent Set in certain graph classes, advancing the understanding of its complexity in structured graphs.

Contribution

It presents new FPT algorithms for MIS in graph classes defined by forbidden subgraphs and noise parameters, resolving the FPT/W[1]-hard dichotomy for five-vertex graphs.

Findings

FPT algorithms for MIS in P(1,t,t,t)-free graphs for fixed t

Randomized FPT algorithms in dart-free and cricket-free graphs

Resolution of the FPT/W[1]-hard dichotomy for five-vertex graphs

Abstract

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 $t \geqslant 1$, 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 $H$.