Parameterized complexity for iterated type partitions and modular-width by Cordasco, Gargano, Rescigno

https://www.doi.org/10.1016/j.dam.2024.03.009

@article{iteratedTypePartitions2024,
author = {Gennaro Cordasco and Luisa Gargano and Adele A. Rescigno},
doi = {10.1016/j.dam.2024.03.009},
issn = {0166-218X},
journaltitle = {Discrete Applied Mathematics},
pages = {100--122},
title = {Parameterized complexity for iterated type partitions and modular-width},
volume = {350},
year = {2024},
}

• page 3 : iterated type partitions – two nodes have the same type iff $N(v) \setminus {u} = N(u) \setminus {v}$ … [ed. paraphrased] let $\mathcal V = {V_1,\dots,V_t}$ be a partition of graph vertices such that each $V_i$ is a clique or an independent set and $t$ is minimized … we can see each element of $\mathcal V$ as a \emph{metavertex} of a new graph $H$, called \emph{type graph} of $G$ … We say that $G$ is a \emph{prime graph} if it matches its type graph … let $H^{(0)}=G$ and $H^{(i)}$ denote the type graph of $H^{(i-1)}$, for $i \ge 1$. Let $d$ be the smallest integer such that $H^{(d)}$ is a \emph{prime graph}. The \emph{iterated type partition} number of $G$, denoted by $\mathrm{itp}(G)$, is the number of nodes of $H^{(d)}$.
• page 3 : neighborhood diversity upper bounds iterated type partitions by a linear function – … $itp(G) \le nd(G)$. Actually $itp(G)$ can be arbitrarily smaller than $nd(G)$.
• page 3 : bounded iterated type partitions does not imply bounded neighborhood diversity – … $itp(G) \le nd(G)$. Actually $itp(G)$ can be arbitrarily smaller than $nd(G)$.
• page 3 : iterated type partitions upper bounds modular-width by a linear function – By definition