On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

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Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the correspond...

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Detalles Bibliográficos
Autores principales:	Bonomo, Flavia, Durán, Guillermo A.
Publicado:	2018
Materias:	(normal, Helly) circular-arc graphs Edge intersection graphs Forbidden induced subgraphs Paths on a grid Powers of cycles Geometry Graphic methods Circular-arc graph Intersection graph Graph theory
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0166218X_v234_n_p12_Alcon http://hdl.handle.net/20.500.12110/paper_0166218X_v234_n_p12_Alcon
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_0166218X_v234_n_p12_Alcon
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spelling	paper:paper_0166218X_v234_n_p12_Alcon2023-06-08T15:15:35Z On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid Bonomo, Flavia Durán, Guillermo A. (normal, Helly) circular-arc graphs Edge intersection graphs Forbidden induced subgraphs Paths on a grid Powers of cycles Geometry Graphic methods Circular-arc graph Forbidden induced subgraphs Intersection graph Paths on a grid Powers of cycles Graph theory Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs. © 2016 Elsevier B.V. Fil:Bonomo, F. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Durán, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2018 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0166218X_v234_n_p12_Alcon http://hdl.handle.net/20.500.12110/paper_0166218X_v234_n_p12_Alcon
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	(normal, Helly) circular-arc graphs Edge intersection graphs Forbidden induced subgraphs Paths on a grid Powers of cycles Geometry Graphic methods Circular-arc graph Forbidden induced subgraphs Intersection graph Paths on a grid Powers of cycles Graph theory
spellingShingle	(normal, Helly) circular-arc graphs Edge intersection graphs Forbidden induced subgraphs Paths on a grid Powers of cycles Geometry Graphic methods Circular-arc graph Forbidden induced subgraphs Intersection graph Paths on a grid Powers of cycles Graph theory Bonomo, Flavia Durán, Guillermo A. On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
topic_facet	(normal, Helly) circular-arc graphs Edge intersection graphs Forbidden induced subgraphs Paths on a grid Powers of cycles Geometry Graphic methods Circular-arc graph Forbidden induced subgraphs Intersection graph Paths on a grid Powers of cycles Graph theory
description	Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs. © 2016 Elsevier B.V.
author	Bonomo, Flavia Durán, Guillermo A.
author_facet	Bonomo, Flavia Durán, Guillermo A.
author_sort	Bonomo, Flavia
title	On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
title_short	On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
title_full	On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
title_fullStr	On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
title_full_unstemmed	On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
title_sort	on the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
publishDate	2018
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0166218X_v234_n_p12_Alcon http://hdl.handle.net/20.500.12110/paper_0166218X_v234_n_p12_Alcon
work_keys_str_mv	AT bonomoflavia onthebendnumberofcirculararcgraphsasedgeintersectiongraphsofpathsonagrid AT duranguillermoa onthebendnumberofcirculararcgraphsasedgeintersectiongraphsofpathsonagrid
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On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

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