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$$ P $$: Maximumb$$ b $$‐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. [ABSTRACT FROM AUTHOR]