On orthogonal ray trees

Mustaţă, Irina; Nishikawa, Kousuke; Takaoka, Asahi; Tayu, Satoshi; Ueno, S.

doi:10.1016/j.dam.2015.07.034

“…Intersection graphs of rays in two directions have been studied by Soto and Telha [30], they show connections with the jump number of some posets and hitting sets of rectangles. The class has been further studied by Shrestha et al [28], and Mustaţȃ et al [22]. The results include polynomial-time recognition and isomorphism algorithms.…”

Section: Previous Work and Motivationmentioning

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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“…The class has been further studied by Shrestha et al [28], and Mustaţȃ et al [22]. The results include polynomial-time recognition and isomorphism algorithms.…”

Section: Previous Work and Motivationmentioning

confidence: 99%

Intersection Graphs of Rays and Grounded Segments

Cardinal

¹

,

Felsner

²

,

Miltzow

³

et al. 2017

Graph-Theoretic Concepts in Computer Science

View full text Add to dashboard Cite

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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“…It is also known that a reflexive digraph is an adjusted interval digraph if and only if it has a min ordering [12]. Moreover, min ordering on bigraphs (bipartite graphs, or more precisely bipartite digraphs with all edges between the parts oriented in the same direction) characterizes the class complements of circular arc graphs [23], and equivalently the class of two-dimensional ray graphs, a natural generalization of interval graphs in the class of bipartite graphs [19,32]. Thus, it has long been believed that digraphs with min ordering are the right notion for the general version of interval digraphs.…”

Section: Introductionmentioning

confidence: 99%

Bi-Arc Digraphs and Conservative Polymorphisms

Hell¹,

Rafiey²,

Rafiey³

2016

Preprint

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We introduce the class of bi-arc digraphs, and show they coincide with the class of digraphs that admit a conservative semi-lattice polymorphism, i.e., a min ordering. Surprisingly this turns out to be also the class of digraphs that admit totally symmetric conservative polymorphisms of all arities. We give an obstruction characterization of, and a polynomial time recognition algorithm for, this class of digraphs. The existence of a polynomial time algorithm was an open problem due to Bagan, Durand, Filiot, and Gauwin.We also discuss a generalization to k-arc digraphs, which has a similar obstruction characterization and recognition algorithm.When restricted to undirected graphs, the class of bi-arc digraphs is included in the previously studied class of bi-arc graphs. In particular, restricted to reflexive graphs, bi-arc digraphs coincide precisely with the well known class of interval graphs. Restricted to reflexive digraphs, they coincide precisely with the class of adjusted interval digraphs, and restricted to bigraphs, they coincide precisely with the class of two directional ray graphs. All these classes have been previously investigated as analogues of interval graphs. We believe that, in a certain sense, bi-arc digraphs are the most general digraph version of interval graphs with nice algorithms and characterizations.

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On orthogonal ray trees

Cited by 3 publications

References 28 publications

Intersection Graphs of Rays and Grounded Segments

Intersection Graphs of Rays and Grounded Segments

Intersection Graphs of Rays and Grounded Segments

Bi-Arc Digraphs and Conservative Polymorphisms

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