Neighborhood covering and independence on $$P_4$$P4-tidy graphs and tree-cographs

Given a simple graph G, a set $$C \subseteq V(G)$$C⊆V(G) is a neighborhood cover set if every edge and vertex of G belongs to some G[v] with $$v \in C$$v∈C, where G[v] denotes the subgraph of G induced by the closed neighborhood of the vertex v. Two elements of $$E(G) \cup V(G)$$E(G)∪V(G) are neighborhood-independent if there is no vertex $$v\in V(G)$$v∈V(G) such that both elements are in G[v]. A set $$S\subseteq V(G)\cup E(G)$$S⊆V(G)∪E(G) is neighborhood-independent if every pair of elements of S is neighborhood-independent. Let $$\rho _{\mathrm {n}}(G)$$ρn(G) be the size of a minimum neighborhood cover set and $$\alpha _{\mathrm {n}}(G)$$αn(G) of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs G as those where the equality $$\rho _{\mathrm {n}}(G^\prime ) = \alpha _{\mathrm {n}}(G^\prime )$$ρn(G′)=αn(G′) holds for every induced subgraph $$G^\prime $$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: $$P_4$$P4-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. Finally we prove that although for complements of trees finding these optimal sets can be achieved in linear-time, for complements of bipartite graphs it is $$\mathrm {NP}$$NP-hard.