On orthogonal ray graphs

Shrestha, Anish Man Singh; Tayu, Satoshi; Ueno, S.

doi:10.1016/j.dam.2010.06.002

Cited by 49 publications

(51 citation statements)

References 19 publications

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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: 98%

Intersection Graphs of Rays and Grounded Segments

Cardinal

Felsner

Miltzow

et al. 2017

Graph-Theoretic Concepts in Computer Science

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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: 98%

Intersection Graphs of Rays and Grounded Segments

Cardinal

Felsner

Miltzow

et al. 2017

Graph-Theoretic Concepts in Computer Science

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“…The class of 2-DORGs has been well studied, and various characterizations are known [50], [52]. Based on a characterization by circular-arc graphs, 2-DORGs can be recognized in O(n 2 ) time, and the orthogonal ray representation of a 2-DORG can be obtained in O(n 2 ) time [50], where n is the number of vertices in the graph. It is known that the biclique problem can be solved in O(n) time provided that orthogonal ray representations are given [49].…”

Section: Introductionmentioning

confidence: 99%

“…The following relationship between bipartite graph classes is known [44], [50]: {Bipartite Permutation Graphs} ⊂ {Biconvex Graphs} ⊂ {Convex Graphs} ⊂ {Interval Bigraphs} ⊂ {2-Directional Orthogonal Ray Graphs} ⊂ {Chordal Bipartite Graphs}, and {2-Directional Orthogonal Ray Graphs} ⊂ {3-Directional Orthogonal Ray Graphs} ⊂ {Orthogonal Ray Graphs} ⊂ {Unit Grid Intersection Graphs} ⊂ {Grid Intersection Graphs}, where X ⊂ Y indicates a set X is a proper subset of Y. Definitions and some comprehensive surveys of these graphs can be found in [10], [30], [50], [53]. The class of 2-DORGs has been well studied, and various characterizations are known [50], [52].…”

Section: Introductionmentioning

confidence: 99%

“…A bipartite graph G with bipartition (U, V) is called an orthogonal ray graph (ORG for short) [50] if there exist a set of disjoint horizontal rays (closed half-lines) R u , u ∈ U, in the xy-plane and a set of disjoint vertical rays R v , v ∈ V, such that for any u ∈ U and v ∈ V, (u, v) ∈ E(G) if and only if R u and R v intersect. Set R(G) = {R w | w ∈ V(G)} is called the orthogonal ray representation of G. ORGs are introduced in connection with the defect-tolerant design of nanocircuits [49].…”

Section: Introductionmentioning

confidence: 99%

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Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

Takaoka

Tayu

Ueno

2014

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Student Member, Satoshi TAYU †b) , Member, and Shuichi UENO †c) , Fellow SUMMARY An orthogonal ray graph is an intersection graph of horizontal and vertical rays (closed half-lines) in the plane. Such a graph is 3-directional if every vertical ray has the same direction, and 2-directional if every vertical ray has the same direction and every horizontal ray has the same direction. We derive some structural properties of orthogonal ray graphs, and based on these properties, we introduce polynomial-time algorithms that solve the dominating set problem, the induced matching problem, and the strong edge coloring problem for these graphs. We show that for 2-directional orthogonal ray graphs, the dominating set problem can be solved in O(n 2 log 5 n) time, the weighted dominating set problem can be solved in O(n 4 log n) time, and the number of dominating sets of a fixed size can be computed in O(n 6 log n) time, where n is the number of vertices in the graph. We also show that for 2-directional orthogonal ray graphs, the weighted induced matching problem and the strong edge coloring problem can be solved in O(n 2 + m log n) time, where m is the number of edges in the graph. Moreover, we show that for 3-directional orthogonal ray graphs, the induced matching problem can be solved in O(m 2 ) time, the weighted induced matching problem can be solved in O(m 4 ) time, and the strong edge coloring problem can be solved in O(m 3 ) time. We finally show that the weighted induced matching problem can be solved in O(m 6 ) time for orthogonal ray graphs.

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“…3. This paper considers the OBDD size of orthogonal ray graphs [24], which have been introduced in connection with the defect-tolerant design of nano-circuits [23], and other related graphs. We show the following:…”

Section: Introductionmentioning

confidence: 99%

OBDD Representation of Intersection Graphs

Takaoka

Tayu

Ueno

2015

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Student Member, Satoshi TAYU †b) , Member, and Shuichi UENO †c) , Fellow SUMMARY Ordered Binary Decision Diagrams (OBDDs for short) are popular dynamic data structures for Boolean functions. In some modern applications, we have to handle such huge graphs that the usual explicit representations by adjacency lists or adjacency matrices are infeasible. To deal with such huge graphs, OBDD-based graph representations and algorithms have been investigated. Although the size of OBDD representations may be large in general, it is known to be small for some special classes of graphs. In this paper, we show upper bounds and lower bounds of the size of OBDDs representing some intersection graphs such as bipartite permu-

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On orthogonal ray graphs

Cited by 49 publications

References 19 publications

Intersection Graphs of Rays and Grounded Segments

Intersection Graphs of Rays and Grounded Segments

Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

OBDD Representation of Intersection Graphs

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