Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás’ Path Argument

Abstract

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S t,t,t as an induced subgraph and provide a subexponential-time algorithm with improved running time \(2^{\mathcal {O}(\sqrt {nt}\log n)}\) and a quasipolynomial-time approximation scheme with improved running time \(2^{\mathcal {O}(\varepsilon ^{-1}t \log ^{5} n)}\) . The Gyárfás’ path argument, a powerful tool that is the main building block for many algorithms in P t -free graphs, ensures that given an n -vertex P t -free graph, in polynomial time we can find a set P of at most t -1 vertices such that every connected component of G-N[P] has at most n /2 vertices. Our main technical contribution is an analog of this result for S t,t,t -free graphs: given an n -vertex S t,t,t -free graph, in polynomial time we can find a set P of \(\mathcal {O}(t \log n)\) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n /2 vertices.