Neighborhood covering and independence on two superclasses of cographs

Given a simple graph G, a set C ⊆ V (G) is a neighborhood cover set if every edge and vertex of G belongs to some G[v] with v ∈ C, where G[v] denotes the subgraph of G induced by the closed neighborhood of the vertex v. Two elements of E(G)∪V (G) are neighborhood-independent if there is no vertex v ∈ V (G) such that both elements are in G[v]. A set S ⊆ V (G) ∪ E(G) is neighborhood-independent if every pair of elements of S is neighborhood-independent. Let ρn(G) be the size of a minimum neighborhood cover set and αn(G) of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs G as those where the equality ρn(G ′) = αn(G ′) holds for every induced subgraph 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: P4-tidy graphs and tree-cographs. We give as well lineartime 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.