Vertex Intersection Graphs of Paths on a Grid

Asinowski, Andrei; Cohen, Elad; Golumbic, Martin Charles; Limouzy, Vincent; Lipshteyn, Marina; Stern, Michal

doi:10.7155/jgaa.00253

Cited by 58 publications

(79 citation statements)

References 20 publications

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“…Our main contribution is that every planar graph is B 2 -VPG (disproving the conjecture of Asinowski et al [3]). This result consists of the following interesting components.…”

Section: Introductionmentioning

confidence: 84%

Planar Graphs as VPG-Graphs

Chaplick

Ueckerdt

2013

Graph Drawing

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A graph is B k -VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs constitute a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n 3/2 ) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we obtain a new proof that bipartite planar graphs are a subclass of 2-DIR.

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“…Our main contribution is that every planar graph is B 2 -VPG (disproving the conjecture of Asinowski et al [3]). This result consists of the following interesting components.…”

Section: Introductionmentioning

confidence: 84%

Planar Graphs as VPG-Graphs

Chaplick

Ueckerdt

2013

Graph Drawing

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“…Asinowski et al [3] showed that the class of VPG-graphs is equivalent to the class of graphs admitting a string-representation. They also defined the class B k -VPG, which contains all VPGgraphs for which each vertex is represented by a path with at most k bends (see [20] for the determination of Copyright © 2018 by SIAM Unauthorized reproduction of this article is prohibited 172 Downloaded 03/24/19 to 52.247.194.177.…”

Section: Introductionmentioning

confidence: 99%

Planar Graphs as L-intersection or L-contact graphs

Isenmann¹,

Pennarun²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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The -intersection graphs are the graphs that have a representation as intersection graphs of axis-parallel shapes in the plane. A subfamily of these graphs are { , |, −}-contact graphs which are the contact graphs of axis parallel , |, and − shapes in the plane. We prove here two results that were conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs areintersection graphs, and that triangle-free planar graphs are { , |, −}-contact graphs. These results are obtained by a new and simple decomposition technique for 4-connected triangulations. Our results also provide a much simpler proof of the known fact that planar graphs are segment intersection graphs. IntroductionThe representation of graphs by contact or intersection of predefined shapes in the plane is a broad subject of research since the work of Koebe on the representation of planar graphs by contacts of circles [28]. In particular, the class of planar graphs has been widely studied in this context.Given a shape 1 X, an X-intersection representation is a collection of X-shaped geometrical objects in the plane. The X-intersection graph described by such a representation has one vertex per geometrical object, and two vertices are adjacent if and only if the corresponding objects intersect. In the case where the shape X defines objects that are homeomorphic to a segment (resp. to a disc), an X-contact representation is an Xintersection representation such that if an intersection occurs between two objects, then it occurs at a single point that is the endpoint of one of them (resp. it occurs on their boundary). We say that a graph G is an X-contact graph if it is the X-intersection graph of an X-contact representation.The case of shapes homeomorphic to discs has * This research is partially supported by the ANR GATO, under contract ANR-16-CE40-0009. 1 We do not provide a formal definition of shape, but a shape characterizes a family of connected geometric objects in the plane. been widely studied; see for example the literature for triangles [19,23], homothetic triangles [25,35], axis parallel rectangles [36], squares [26,33], hexagons [22], convex bodies [34], or axis aligned polygons [2]. Here, we focus on intersection and contact representations of planar graphs with objects that are homeomorphic to a segment. The more general representations of this type are the intersection or contact representation with curves. those are called string representations. It is known that every planar graph has a stringintersection representation [28]. However, if one forbids tangent curves, this representation may contain pairs of curves that cross several times. One may thus take an additional parameter into account, namely the maximal number of crossings of any two of the curves: a 1-string representation of a graph is a string representation where every two curves intersect at most once. The question of finding a 1-string representation of planar graphs has been solved by Chalopin et al. in the positive [11], and additional parameters are now ...

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“…A graph is said to be simply a VPG graph if it is a B k -VPG graph for some k. Several relationships between VPG graphs and other known graph classes have been studied in [ACG + 11]. VPG graphs are known to be equivalent to the known class of string graphs, which are the intersection graphs of simple curves in the plane [ACG + 12]. B 0 -VPG graphs are a special case of segment graphs (intersection graphs of line segments in the plane), which are known as 2-DIR graphs.…”

Section: Introductionmentioning

confidence: 99%

VPG and EPG bend-numbers of Halin graphs

Francis

Lahiri

2016

Discrete Applied Mathematics

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A piecewise linear simple curve in the plane made up of k + 1 line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a k-bend path. Given a graph G, a collection of k-bend paths in which each path corresponds to a vertex in G and two paths have a common point if and only if the vertices corresponding to them are adjacent in G is called a B k -VPG representation of G. Similarly, a collection of k-bend paths each of which corresponds to a vertex in G is called an B k -EPG representation of G if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in G. The VPG bend-number b v (G) of a graph G is the minimum k such that G has a B k -VPG representation. Similarly, the EPG bend-number b e (G) of a graph G is the minimum k such that G has a B k -EPG representation. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph then b v (G) ≤ 1 and b e (G) ≤ 2. These bounds are tight. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths.

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Vertex Intersection Graphs of Paths on a Grid

Cited by 58 publications

References 20 publications

Planar Graphs as VPG-Graphs

Planar Graphs as VPG-Graphs

Planar Graphs as L-intersection or L-contact graphs

VPG and EPG bend-numbers of Halin graphs

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