Perfect edge domination: hard and solvable cases

Let G be an undirected graph. An edge of Gdominates itself and all edges adjacent to it. A subset E′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E′. We say that E′ is a perfect edge dominating set of G, if every edge not in E′ is dominated by...

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Guardado en:
Detalles Bibliográficos
Publicado: 2018
Materias:
Claw-free graphs
Complexity dichotomy
Cubic graphs
NP-completeness
Perfect edge domination
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02545330_v264_n1-2_p287_Lin
http://hdl.handle.net/20.500.12110/paper_02545330_v264_n1-2_p287_Lin
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_02545330_v264_n1-2_p287_Lin
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spelling paper:paper_02545330_v264_n1-2_p287_Lin2023-06-08T15:22:01Z Perfect edge domination: hard and solvable cases Claw-free graphs Complexity dichotomy Cubic graphs NP-completeness Perfect edge domination Let G be an undirected graph. An edge of Gdominates itself and all edges adjacent to it. A subset E′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E′. We say that E′ is a perfect edge dominating set of G, if every edge not in E′ is dominated by exactly one edge of E′. The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most d≥ 3 and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a P5-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a P5. © 2017, Springer Science+Business Media, LLC. 2018 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02545330_v264_n1-2_p287_Lin http://hdl.handle.net/20.500.12110/paper_02545330_v264_n1-2_p287_Lin
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Claw-free graphs
Complexity dichotomy
Cubic graphs
NP-completeness
Perfect edge domination
spellingShingle Claw-free graphs
Complexity dichotomy
Cubic graphs
NP-completeness
Perfect edge domination
Perfect edge domination: hard and solvable cases
topic_facet Claw-free graphs
Complexity dichotomy
Cubic graphs
NP-completeness
Perfect edge domination
description Let G be an undirected graph. An edge of Gdominates itself and all edges adjacent to it. A subset E′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E′. We say that E′ is a perfect edge dominating set of G, if every edge not in E′ is dominated by exactly one edge of E′. The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most d≥ 3 and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a P5-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a P5. © 2017, Springer Science+Business Media, LLC.
title Perfect edge domination: hard and solvable cases
title_short Perfect edge domination: hard and solvable cases
title_full Perfect edge domination: hard and solvable cases
title_fullStr Perfect edge domination: hard and solvable cases
title_full_unstemmed Perfect edge domination: hard and solvable cases
title_sort perfect edge domination: hard and solvable cases
publishDate 2018
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02545330_v264_n1-2_p287_Lin
http://hdl.handle.net/20.500.12110/paper_02545330_v264_n1-2_p287_Lin
_version_ 1768542643418038272

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