On the bend-number of planar and outerplanar graphs

Heldt, Daniel; Knauer, Kolja; Ueckerdt, Torsten

doi:10.1016/j.dam.2014.07.015

Cited by 32 publications

(37 citation statements)

References 29 publications

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“…The bend number of classical graph classes was investigated as well. In [14], it was shown that outerplanar graphs are B 2 -EPG graphs and that planar graphs are B 4 -EPG graphs. For planar graphs, it is still an open question whether their bend number is equal to 3 or 4.…”

Section: Introductionmentioning

confidence: 99%

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

Alcón

Bonomo

Durán

et al. 2018

Discrete Applied Mathematics

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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 corresponding paths share at least one edge of the grid. For a nonnegative integer k, B k -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 B 4 -EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is B 3 -EPG, and that there exist circular-arc graphs which are not B 2 -EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle E-mail addresses: of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define B k -EPR graphs, k ≥ 0, the same way as B k -EPG graphs. Circulararc graphs are clearly B 4 -EPR graphs and we will show that there exist circular-arc graphs that are not B 3 -EPR graphs. We also show that normal circular-arc graphs are B 2 -EPR graphs and that there exist normal circular-arc graphs that are not B 1 -EPR graphs. Finally, we characterize B 1 -EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

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Section: Introductionmentioning

confidence: 99%

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

Alcón

Bonomo

Durán

et al. 2018

Discrete Applied Mathematics

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“…90 degrees turns at a grid-point) that the paths may have: a graph is a B k -VPG graph, for some integer k ≥ 0, if one can assign a path on a grid having at most k bends to each vertex such that two vertices are adjacent if and only if the corresponding paths intersect on at least one grid-point. Since their introduction, B k -VPG graphs have received much attention (see for instance [3,4,[8][9][10][11]17,19,21,22,24]).…”

Section: Introductionmentioning

confidence: 99%

Characterising circular-arc contact B0–VPG graphs

Bonomo

Galby

Gonzalez

2020

Discrete Applied Mathematics

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A contact B 0 -VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-toone correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown in [15] that Recognition is NP-complete for contact B 0 -VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact B 0 -VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs.

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“…90 degrees turns at a grid-point) that a path may have; for k ≥ 0, the class B k -EPG consists of those EPG graphs admitting a representation in which each path has at most k bends. Since their introduction, B k -EPG graphs have been extensively studied from several points of view (see for instance [1,2,3,5,6,7,8,9,14,15]). One major interest is the so-called bend number ; for a graph class G, the bend number of G is the minimum integer k ≥ 0 such that every graph G ∈ G is a B k -EPG graph.…”

Section: Introductionmentioning

confidence: 99%