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

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S-t,S- t,S- t as an induced subgraph and provide a subexponential-time algorithm with improved running time 2(O(root nt log n)) and a quasipolynomial-time approximation scheme with improved running time 2(O(epsilon-1 t log5 n)). The Gyarfas' 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,S- t,S- t-free graphs: given an n-vertex S-t,S- t,S- t-free graph, in polynomial time we can find a set P of O(t logn) 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.