Intersection Dimension of Bipartite Graphs

Chaplick, Steven; Hell, Pavol; Otachi, Yota; Saitoh, Toshiki; Uehara, Ryuhei

doi:10.1007/978-3-319-06089-7_23

Cited by 9 publications

(4 citation statements)

References 31 publications

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“…For these results we note that graphs with a special SegRay representation have dimension at most 3 and that the interval dimension of SegRay graphs is bounded by 3. This latter result was shown previously by a different argument in [6].…”

Section: Dimensionsupporting

confidence: 87%

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Grid Intersection Graphs and Order Dimension

Chaplick

Felsner

Hoffmann

et al. 2018

Order

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We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we provide a comprehensive study of classes of graphs between grid intersection graphs and bipartite permutation graphs and the containment relation on these classes. Order dimension plays a role in many arguments.

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Section: Dimensionsupporting

confidence: 87%

“…First, we will show that the dimension of GIGs is bounded. It has previously been observed that idim(G) ≤ 4 when G is a GIG [6]. As already shown in [13] this can be strengthened to dim(G) ≤ 4.…”

Section: Dimensionsupporting

confidence: 61%

Grid Intersection Graphs and Order Dimension

Chaplick

Felsner

Hoffmann

et al. 2018

Order

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“…They introduced several classes of intersection graphs that are the topic of this paper. Geometric intersection graphs are now ubiquitous in discrete and computational geometry, and deep connections to other fields such as complexity theory [20,25,26] and order dimension theory [8,9,11] have been established.…”

Section: Introductionmentioning

confidence: 99%

Intersection Graphs of Rays and Grounded Segments

Cardinal¹,

Felsner²,

Miltzow³

et al. 2018

JGAA

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We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that:• intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class,• not every intersection graph of rays is an intersection graph of downward rays, and• not every intersection graph of rays is an outer segment graph.The first result answers an open problem posed by Cabello and Jejčič. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.

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“…If the horizontal elements are segments and the vertical elements are rays, then the corresponding intersection graphs are called SegRay graphs [7][8][9]22]. A graph G is a SegRay graph if and only if M (G) can be permuted to avoid γ 4 [10].…”

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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Intersection Dimension of Bipartite Graphs

Cited by 9 publications

References 31 publications

Grid Intersection Graphs and Order Dimension

Grid Intersection Graphs and Order Dimension

Intersection Graphs of Rays and Grounded Segments

Recognition and Drawing of Stick Graphs

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