Abstract:
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.
Registro:
Documento: |
Artículo
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Título: | On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid |
Autor: | Alcón, L.; Bonomo, F.; Durán, G.; Gutierrez, M.; Mazzoleni, M.P.; Ries, B.; Valencia-Pabon, M. |
Filiación: | Dto. de Matemática, FCE-UNLP, La Plata, Argentina Dto. de Computación FCEN-UBA, Buenos Aires, Argentina Dto. de Matemática e Inst. de Cálculo FCEN-UBA, Buenos Aires, Argentina Dto. de Ingeniería Industrial, FCFM-Univ. de Chile, Santiago, Chile Université de Fribourg, DIUF, Fribourg, Switzerland Université Paris-13, Sorbonne Paris Cité LIPN, CNRS UMR7030, Villetaneuse, France CONICET, Argentina Délégation at the INRIA Nancy - Grand Est, France
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Palabras clave: | (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 |
Año: | 2018
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Volumen: | 234
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Página de inicio: | 12
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Página de fin: | 21
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DOI: |
http://dx.doi.org/10.1016/j.dam.2016.08.004 |
Título revista: | Discrete Applied Mathematics
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Título revista abreviado: | Discrete Appl Math
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ISSN: | 0166218X
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CODEN: | DAMAD
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0166218X_v234_n_p12_Alcon |
Referencias:
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- Asinowski, A., Ries, B., Some properties of edge intersection graphs of single-bend paths on a grid (2012) Discrete Math., 312, pp. 427-440
- Asinowski, A., Suk, A., Edge intersection graphs of systems of paths on a grid with a bounded number of bends (2009) Discrete Appl. Math., 157, pp. 3174-3180
- Biedl, T., Stern, M., On edge intersection graphs of k-bend paths in grids (2010) Discrete Math. Theor. Comput. Sci., 12 (1), pp. 1-12
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Citas:
---------- APA ----------
Alcón, L., Bonomo, F., Durán, G., Gutierrez, M., Mazzoleni, M.P., Ries, B. & Valencia-Pabon, M.
(2018)
. On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid. Discrete Applied Mathematics, 234, 12-21.
http://dx.doi.org/10.1016/j.dam.2016.08.004---------- CHICAGO ----------
Alcón, L., Bonomo, F., Durán, G., Gutierrez, M., Mazzoleni, M.P., Ries, B., et al.
"On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid"
. Discrete Applied Mathematics 234
(2018) : 12-21.
http://dx.doi.org/10.1016/j.dam.2016.08.004---------- MLA ----------
Alcón, L., Bonomo, F., Durán, G., Gutierrez, M., Mazzoleni, M.P., Ries, B., et al.
"On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid"
. Discrete Applied Mathematics, vol. 234, 2018, pp. 12-21.
http://dx.doi.org/10.1016/j.dam.2016.08.004---------- VANCOUVER ----------
Alcón, L., Bonomo, F., Durán, G., Gutierrez, M., Mazzoleni, M.P., Ries, B., et al. On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid. Discrete Appl Math. 2018;234:12-21.
http://dx.doi.org/10.1016/j.dam.2016.08.004