Getting linear time in graphs of bounded neighborhood diversity

Parameterized complexity, introduced to efficiently solve NP-hard problems for small values of a fixed parameter, has been recently used as a tool to speed up algorithms for tractable problems. Following this line of research, we design algorithms parameterized by neighborhood diversity (nd$$ \mathsf{nd} $$) for several graph theoretic problems in P$$ P $$: Maximum b$$ b $$-Matching, Triangle Counting and Listing, Girth, Global Minimum Vertex Cut, and Perfect Graphs Recognition. Such problems are known to admit algorithms parameterized by modular-width (mw$$ \mathsf{mw} $$) and consequently-as nd$$ \mathsf{nd} $$ is a special case of mw$$ \mathsf{mw} $$-by nd$$ \mathsf{nd} $$. However, the proposed novel algorithms allow for improving the computational complexity from time O(f(mw)n+m)$$ O\left(f\left(\mathsf{mw}\right)\cdotp n+m\right) $$-where n$$ n $$ and m$$ m $$ denote, respectively, the number of vertices and edges in the input graph-to time O(g(nd)+n+m)$$ O\left(g\left(\mathsf{nd}\right)+n+m\right) $$ which is only additive in the size of the input. Then we consider some classical NP-hard problems (Maximum independent set, Maximum clique, and Minimum dominating set) and show that for several classes of hereditary graphs, they admit linear time algorithms for sufficiently small-nonnecessarily constant-values of the neighborhood diversity parameter.