On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
- Alcón, Liliana National University of La Plata
- Bonomo, Flavia University of Buenos Aires, CONICET
- Durán, Guillermo University of Buenos Aires, University of Chile, CONICET
- Gutierrez, Marisa National University of La Plata, CONICET
- Mazzoleni, María Pía National University of La Plata, CONICET
- Ries, Bernard University of Fribourg
- Valencia-Pabon, Mario Université Paris-13
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2018
Published in:
- Discrete applied mathematics. - 2018, vol. 234, p. 12-21
Edge intersection graphs
paths on a grid
forbidden induced subgraphs
(normal, Helly) circular-arc graphs
powers of cycles
English
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 circulararc 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.
- Faculty
- Faculté des sciences économiques et sociales et du management
- Department
- Département d'informatique
- Language
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- English
- Classification
- Economics
- License
- License undefined
- Identifiers
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- RERO DOC 324551
- DOI 10.1016/j.dam.2016.08.004
- Persistent URL
- https://folia.unifr.ch/unifr/documents/307803
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