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

DURÁN, GUILLERMO; SAFE, MARTÍN; WARNES, XAVIER

ANNALS OF OPERATIONS RESEARCH

SPRINGER

Año: 2020 vol. 286 p. 55 - 86

0254-5330

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 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 -hard.