Jump Number of Two-Directional Orthogonal Ray Graphs

Soto, José A.; Telha, Claudio

doi:10.1007/978-3-642-20807-2_31

Cited by 18 publications

(21 citation statements)

References 25 publications

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“…Let (x v , y v ) be the endpoint of R v ∈ R(G), and we assume without loss of generality that the x-coordinates are distinct and the y-coordinates are distinct [26]. Notice that for any u ∈ U and w ∈ W, (u, w) ∈ E(G) if and only if x u < x w and y u < y w .…”

Section: Two-directional Orthogonal Ray Graphsmentioning

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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Section: Two-directional Orthogonal Ray Graphsmentioning

confidence: 99%

OBDD Representation of Intersection Graphs

Takaoka

Tayu

Ueno

2015

IEICE Trans. Inf. & Syst.

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“…It is shown in [51], [52] that a bipartite graph G with bipartition (U, V) is a 2-DORG if and only if there exists a set of points p w , w ∈ V(G), in the xy-plane such that for any u ∈ U and v ∈ V, (u, v) ∈ E(G) if and only if p u < R 2 p v . Moreover, we can assume without loss of generality that every x w is distinct and every y w is distinct.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let G be a 2-DORG with bipartition (U, V) and a point representation P(G) = {(x w , y w ) | w ∈ V(G)}. We assume without loss of generality that every x w is distinct and every y w is distinct [52]. For each u ∈ U, let B u be a square with ur(B u ) = (x u , y u ) and ll (B u Fig.…”

Section: Dominating Set Problemmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…It is known that the biclique problem can be solved in O(n) time provided that orthogonal ray representations are given [49]. It is also known that the cross-free matching problem, the biclique cover problem, and the jump number problem can be solved in O(n 2.5 log c n) time for some constant c [52]. Based on a characterization by vertex ordering, the matching problem and the oriented coloring problem can be solved in linear time [22].…”

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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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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Jump Number of Two-Directional Orthogonal Ray Graphs

Cited by 18 publications

References 25 publications

OBDD Representation of Intersection Graphs

OBDD Representation of Intersection Graphs

Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

Recognition and Drawing of Stick Graphs

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