Total dominating sequences in trees, split graphs, and under modular decomposition

Autores
Brešar, Boštjan; Kos, Tim; Nasini, Graciela Leonor; Torres, Pablo Daniel
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.
Fil: Brešar, Boštjan. University of Maribor; Eslovenia. Institute of Mathematics, Physics and Mechanics; Eslovenia
Fil: Kos, Tim. Institute of Mathematics, Physics and Mechanics; Eslovenia
Fil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Materia
GRUNDY TOTAL DOMINATION NUMBER
MODULAR DECOMPOSITION
SPLIT GRAPH
TREE
VERTEX COVER
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/117720
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/117720
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Total dominating sequences in trees, split graphs, and under modular decompositionBrešar, BoštjanKos, TimNasini, Graciela LeonorTorres, Pablo DanielGRUNDY TOTAL DOMINATION NUMBERMODULAR DECOMPOSITIONSPLIT GRAPHTREEVERTEX COVERhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.Fil: Brešar, Boštjan. University of Maribor; Eslovenia. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Kos, Tim. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaElsevier Science2018-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/117720Brešar, Boštjan; Kos, Tim; Nasini, Graciela Leonor; Torres, Pablo Daniel; Total dominating sequences in trees, split graphs, and under modular decomposition; Elsevier Science; Discrete Optimization; 28; 5-2018; 16-301572-5286CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.disopt.2017.10.002info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S1572528617302293info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1608.06804info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:15:17Zoai:ri.conicet.gov.ar:11336/117720instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:15:17.393CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Total dominating sequences in trees, split graphs, and under modular decomposition
title Total dominating sequences in trees, split graphs, and under modular decomposition
spellingShingle Total dominating sequences in trees, split graphs, and under modular decomposition
Brešar, Boštjan
GRUNDY TOTAL DOMINATION NUMBER
MODULAR DECOMPOSITION
SPLIT GRAPH
TREE
VERTEX COVER
title_short Total dominating sequences in trees, split graphs, and under modular decomposition
title_full Total dominating sequences in trees, split graphs, and under modular decomposition
title_fullStr Total dominating sequences in trees, split graphs, and under modular decomposition
title_full_unstemmed Total dominating sequences in trees, split graphs, and under modular decomposition
title_sort Total dominating sequences in trees, split graphs, and under modular decomposition
dc.creator.none.fl_str_mv Brešar, Boštjan
Kos, Tim
Nasini, Graciela Leonor
Torres, Pablo Daniel
author Brešar, Boštjan
author_facet Brešar, Boštjan
Kos, Tim
Nasini, Graciela Leonor
Torres, Pablo Daniel
author_role author
author2 Kos, Tim
Nasini, Graciela Leonor
Torres, Pablo Daniel
author2_role author
author
author
dc.subject.none.fl_str_mv GRUNDY TOTAL DOMINATION NUMBER
MODULAR DECOMPOSITION
SPLIT GRAPH
TREE
VERTEX COVER
topic GRUNDY TOTAL DOMINATION NUMBER
MODULAR DECOMPOSITION
SPLIT GRAPH
TREE
VERTEX COVER
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.
Fil: Brešar, Boštjan. University of Maribor; Eslovenia. Institute of Mathematics, Physics and Mechanics; Eslovenia
Fil: Kos, Tim. Institute of Mathematics, Physics and Mechanics; Eslovenia
Fil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
description A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.
publishDate 2018
dc.date.none.fl_str_mv 2018-05
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/117720
Brešar, Boštjan; Kos, Tim; Nasini, Graciela Leonor; Torres, Pablo Daniel; Total dominating sequences in trees, split graphs, and under modular decomposition; Elsevier Science; Discrete Optimization; 28; 5-2018; 16-30
1572-5286
CONICET Digital
CONICET
url http://hdl.handle.net/11336/117720
identifier_str_mv Brešar, Boštjan; Kos, Tim; Nasini, Graciela Leonor; Torres, Pablo Daniel; Total dominating sequences in trees, split graphs, and under modular decomposition; Elsevier Science; Discrete Optimization; 28; 5-2018; 16-30
1572-5286
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.disopt.2017.10.002
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S1572528617302293
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1608.06804
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier Science
publisher.none.fl_str_mv Elsevier Science
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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