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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Autores principales: Alcón, L., Bonomo, F., Durán, G., Gutierrez, M., Mazzoleni, M.P., Ries, B., Valencia-Pabon, M.
Formato: JOUR
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:http://hdl.handle.net/20.500.12110/paper_0166218X_v234_n_p12_Alcon
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_0166218X_v234_n_p12_Alcon2023-10-03T15:03:44Z On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid Alcón, L. Bonomo, F. Durán, G. Gutierrez, M. Mazzoleni, M.P. Ries, B. Valencia-Pabon, M. (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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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
Alcón, L.
Bonomo, F.
Durán, G.
Gutierrez, M.
Mazzoleni, M.P.
Ries, B.
Valencia-Pabon, M.
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.
format JOUR
author Alcón, L.
Bonomo, F.
Durán, G.
Gutierrez, M.
Mazzoleni, M.P.
Ries, B.
Valencia-Pabon, M.
author_facet Alcón, L.
Bonomo, F.
Durán, G.
Gutierrez, M.
Mazzoleni, M.P.
Ries, B.
Valencia-Pabon, M.
author_sort Alcón, L.
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
url http://hdl.handle.net/20.500.12110/paper_0166218X_v234_n_p12_Alcon
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