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Given a simple graph $G$, a set $C \subseteq V(G)$ is a neighborhood coverset 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 ofthe vertex $v$. Two elements of $E(G) \cup V(G)$ are neighborhood-independentif 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 ofelements of $S$ is neighborhood-independent. Let $\rho_{\mathrm n}(G)$ be thesize of a minimum neighborhood cover set and $\alpha_{\mathrm n}(G)$ of amaximum neighborhood-independent set. Lehel and Tuza definedneighborhood-perfect graphs $G$ as those where the equality $\rho_{\mathrmn}(G') = \alpha_{\mathrm n}(G')$ holds for every induced subgraph $G'$ of $G$.  In this work we prove forbidden induced subgraph characterizations of theclass of neighborhood-perfect graphs, restricted to two superclasses ofcographs: $P_4$-tidy graphs and tree-cographs. We give as well linear-timealgorithms for solving the recognition problem of neighborhood-perfect graphsand the problem of finding a minimum neighborhood cover set and a maximumneighborhood-independent set in these same classes.

Neighborhood covering and independence on two superclasses of cographs