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

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-...

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Detalles Bibliográficos
Autores principales:	Durán, G., Safe, M., Warnes, X.
Formato:	INPR
Materias:	$$P_4$$P4-tidy graphs Co-bipartite graphs Forbidden induced subgraphs Neighborhood-perfect graphs Recognition algorithms Tree-cographs
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_02545330_v_n_p1_Duran
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

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spelling	todo:paper_02545330_v_n_p1_Duran2023-10-03T15:11:36Z Neighborhood covering and independence on P4-tidy graphs and tree-cographs Durán, G. Safe, M. Warnes, X. $$P_4$$P4-tidy graphs Co-bipartite graphs Forbidden induced subgraphs Neighborhood-perfect graphs Recognition algorithms Tree-cographs 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. © 2017 Springer Science+Business Media, LLC, part of Springer Nature INPR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_02545330_v_n_p1_Duran
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	$$P_4$$P4-tidy graphs Co-bipartite graphs Forbidden induced subgraphs Neighborhood-perfect graphs Recognition algorithms Tree-cographs
spellingShingle	$$P_4$$P4-tidy graphs Co-bipartite graphs Forbidden induced subgraphs Neighborhood-perfect graphs Recognition algorithms Tree-cographs Durán, G. Safe, M. Warnes, X. Neighborhood covering and independence on P4-tidy graphs and tree-cographs
topic_facet	$$P_4$$P4-tidy graphs Co-bipartite graphs Forbidden induced subgraphs Neighborhood-perfect graphs Recognition algorithms Tree-cographs
description	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. © 2017 Springer Science+Business Media, LLC, part of Springer Nature
format	INPR
author	Durán, G. Safe, M. Warnes, X.
author_facet	Durán, G. Safe, M. Warnes, X.
author_sort	Durán, G.
title	Neighborhood covering and independence on P4-tidy graphs and tree-cographs
title_short	Neighborhood covering and independence on P4-tidy graphs and tree-cographs
title_full	Neighborhood covering and independence on P4-tidy graphs and tree-cographs
title_fullStr	Neighborhood covering and independence on P4-tidy graphs and tree-cographs
title_full_unstemmed	Neighborhood covering and independence on P4-tidy graphs and tree-cographs
title_sort	neighborhood covering and independence on p4-tidy graphs and tree-cographs
url	http://hdl.handle.net/20.500.12110/paper_02545330_v_n_p1_Duran
work_keys_str_mv	AT durang neighborhoodcoveringandindependenceonp4tidygraphsandtreecographs AT safem neighborhoodcoveringandindependenceonp4tidygraphsandtreecographs AT warnesx neighborhoodcoveringandindependenceonp4tidygraphsandtreecographs
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Neighborhood covering and independence on P4-tidy graphs and tree-cographs

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