Max Weight Independent Set in graphs with no long claws: An analog of the Gyárfás' path argument
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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 [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time 2^𝒪(√(n)log n) and a quasipolynomial-time approximation scheme with improved running time 2^𝒪(ε^-1log^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 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.
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