Polynomial Turing Compressions for Some Graph Problems Parameterized by Modular-Width

In this paper we investigate the parameterized complexity for NP-hard graph problems parameterized by a structural parameter modular-width. We develop a recipe that is able to simplify the process of obtaining polynomial Turing compressions for a class of graph problems parameterized by modular-width. Moreover, we prove that several problems, which include \textsc{chromatic number, independent set}, \textsc{Hamiltonian cycle}, etc. have polynomial Turing compressions parameterized by modular-width. In addition, under the assumption that P $\neq$ NP, we provide tight kernels for a few problems such as \textsc{Steiner tree} parameterized by modular-width. Meanwhile, we demonstrate that some problems, which includes \textsc{dominating set}, \textsc{odd cycle transversal}, \textsc{connected vertex cover}, etc. are fixed-parameter tractable parameterized by modular-width.