Grid Intersection Graphs and Order Dimension

Chaplick, Steven; Felsner, Stefan; Hoffmann, Udo; Wiechert, Veit

doi:10.1007/s11083-017-9437-0

Cited by 16 publications

(22 citation statements)

References 30 publications

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“…. , u 6 adjacent to v k in G k are represented by disks {D 2, j } j∈ [6] that must all intersectD 1 and must not intersect among themselves. Therefore, by Observation 7, at least one of them, let's call itD 2 , is of a size strictly smaller thanD 1 .…”

Section: Disk Graphsmentioning

confidence: 99%

“…Therefore, by Observation 7, at least one of them, let's call itD 2 , is of a size strictly smaller thanD 1 . We repeat the same argument at an arbitrary "level" i; at least one of the disks {D i, j } j∈ [6] must be of a strictly smaller size thanD i−1 in order to be able to intersect it, as well as to avoid intersecting among themselves. We denote it byD i .…”

Section: Disk Graphsmentioning

confidence: 99%

“…Janson and Kratochvíl [8] analyse the probability that a random graph is some type of a geometric intersection graph. In their recent work, Chaplick et al [6] also study the strict inclusion between different types of geometric intersection graphs with similar definitions.…”

Section: Introductionmentioning

confidence: 99%

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Refining the Hierarchies of Classes of Geometric Intersection Graphs

Cabello

Jejčič

2017

Electron. J. Combin.

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We analyse properties of geometric intersection graphs to show strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties:• A graph G is outerplanar if and only if the 1-subdivision of G is outer-segment.• For each integer k ≥ 1, the class of intersection graphs of segments with k different lengths is a strict subclass of the class of intersection graphs of segments with k + 1 different lengths.• For each integer k ≥ 1, the class of intersection graphs of disks with k different sizes is a strict subclass of the class of intersection graphs of disks with k + 1 different sizes.• The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

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Section: Disk Graphsmentioning

confidence: 99%

Section: Disk Graphsmentioning

confidence: 99%

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Refining the Hierarchies of Classes of Geometric Intersection Graphs

Cabello

Jejčič

2017

Electron. J. Combin.

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“…The recognition problem is NP-complete for both GIG [24] and UGIG [26]. We note that 4DOR is a subset of UGIG but Stick is not [8].…”

Section: Introductionmentioning

confidence: 99%

“…Hook graphs are also known as max point-tolerance graphs [5] and heterozygosity graphs [17]. The bipartite graphs that admit a Hook representation are called BipHook [8]. The complexities of recognizing the classes StabGIG, BipHook, and Stick are all open [8].…”

Section: Introductionmentioning

confidence: 99%

Recognition and Drawing of Stick Graphs

Luca

Hossain

Kobourov

et al. 2018

Lecture Notes in Computer Science

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A Stick graph is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a "ground line," a line with slope −1.It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G's bipartite adjacency matrix with rows A and columns B excludes three small 'forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A| 3 |B| 3 )-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.

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Stick Graphs with Length Constraints

Chaplick

Kindermann

Löffler

et al. 2019

Lecture Notes in Computer Science

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Stick graphs are intersection graphs of horizontal and vertical line segments that all touch a line of slope −1 and lie above this line. De Luca et al. [GD'18] considered the recognition problem of stick graphs when no order is given (STICK), when the order of either one of the two sets is given (STICK A ), and when the order of both sets is given (STICK AB ). They showed how to solve STICK AB efficiently. In this paper, we improve the running time of their algorithm, and we solve STICK A efficiently. Further, we consider variants of these problems where the lengths of the sticks are given as input. We show that these variants of STICK, STICK A , and STICK AB are all NP-complete. On the positive side, we give an efficient solution for STICK AB with fixed stick lengths if there are no isolated vertices.

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Grid Intersection Graphs and Order Dimension

Cited by 16 publications

References 30 publications

Refining the Hierarchies of Classes of Geometric Intersection Graphs

Refining the Hierarchies of Classes of Geometric Intersection Graphs

Recognition and Drawing of Stick Graphs

Stick Graphs with Length Constraints

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