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

Abstract

Given a simple graph G, a set (Formula presented.) is a neighborhood cover set if every edge and vertex of G belongs to some G[v] with (Formula presented.), where G[v] denotes the subgraph of G induced by the closed neighborhood of the vertex v. Two elements of (Formula presented.) are neighborhood-independent if there is no vertex (Formula presented.) such that both elements are in G[v]. A set (Formula presented.) is neighborhood-independent if every pair of elements of S is neighborhood-independent. Let (Formula presented.) be the size of a minimum neighborhood cover set and (Formula presented.) of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs G as those where the equality (Formula presented.) holds for every induced subgraph (Formula presented.) of G. In this work we prove forbidden induced subgraph characterizations of the class of neighborhood-perfect graphs, restricted to two superclasses of cographs: (Formula presented.)-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 (Formula presented.)-hard.