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

Abstract

Given a simple graph G, a set C subset of V(G)\documentclass[12pt] is a neighborhood cover set if every edge and vertex of G belongs to some G[v] with v is an element of C\documentclass[12pt] denotes the subgraph of G induced by the closed neighborhood of the vertex v. Two elements of E(G)?V(G)\documentclass[12pt] are neighborhood-independent if there is no vertex v is an element of V(G)\documentclass[12pt]such that both elements are in G[v]. A set S subset of V(G)?E(G)\documentclass[12pt] is neighborhood-independent if every pair of elements of S is neighborhood-independent. Let rho n(G)\documentclass[12pt] be the size of a minimum neighborhood cover set and alpha n(G)\documentclass[12pt] of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs G as those where the equality rho n(G ')=alpha n(G ')\documentclass[12pt] holds for every induced subgraph G-documentclass[12pt] 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\documentclass[12pt]-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 NP\documentclass[12pt]-hard.