Neighborhood covering and independence on two superclasses of cographs

Given a simple graph $G$, a set $C\subseteq V(G)$ is a neighborhood cover set if every edge and vertex of $G$ belongs to some $G[v]$ with $v\in C$, where $G[v]$ denotes the subgraph of $G$ induced by the closed neighborhood of the vertex $v$. Two elements of $E(G)\cup V(G)$ are neighborhood-independent if there is no vertex $v\in V(G)$ such that both elements are in $G[v]$. A set $S\subseteq V(G)\cup E(G)$ is neighborhood-independent if every pair of elements of $S$ is neighborhood-independent. Let ${\rho }_{\mathrm{n}}(G)$ be the size of a minimum neighborhood cover set and ${\alpha }_{\mathrm{n}}(G)$ of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs $G$ as those where the equality $\rho_{\mathrm n}(G’) = \alpha_{\mathrm n}(G’)$holdsforeveryinducedsubgraph$G’$of$G$. In this work we prove forbidden induced subgraph characterizations of the class of neighborhood-perfect graphs, restricted to two superclasses of cographs: $P_4$-tidy graphs and tree-cographs. We give as well linear-time algorithms for solving the recognition problem of neighborhood-perfect graphs and the problem of finding a minimum neighborhood cover set and a maximum neighborhood-independent set in these same classes.