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

Abstract

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 $\rho _{\mathrm {n}}(G^\prime ) = \alpha _{\mathrm {n}}(G^\prime )$ holds for every induced subgraph $G^\prime$ 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. 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 $\mathrm {NP}$-hard.