Orthogonal ray graphs and nano-PLA design

“…To see that every 4-DORG is a UGIG, we first fix an appropriate length for the segments, e.g., the length d of the diagonal of R. If we only keep the initial part of length d from each ray we get a UGIG representation. Essentially this construction was already used in [18].…”

Section: -Dorgs and Ugigsmentioning

confidence: 99%

“…4-DORGs in VLSI design. In [18] 4-DORGs were introduced as a mathematical model for defective nano-crossbars in PLA (programmable logic arrays) design. A nano-crossbar is a rectangular circuit board with m × n orthogonally crossing wires.…”

mentioning

confidence: 99%

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On the Recognition of Four-Directional Orthogonal Ray Graphs

Felsner

Mertzios

Musta

2013

Mathematical Foundations of Computer Science 2013

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Abstract. A 4-directional orthogonal ray graph (4-DORG) is the intersection graph of horizontal and vertical rays. If the rays are only pointing into the positive x and y directions, the intersection graph is a 2-DORG. For 3-DORGs horizontal rays are unrestricted but vertical rays only use the positive direction. The recognition of 2-DORGs is known to be polynomial, they form a nice subclass of bipartite comparability graphs. The recognition problems for 3-DORGs and 4-DORGs, however, are open. Recently is has been shown that the recognition of unit grid intersection graphs, a superclass of 4-DORGs, is NP-complete. Suppose G is given with a partition {L, R, U, D} of its vertices and the question is whether G has a 4-DORG representation, where the four classes of the partition correspond to the four directions of rays. We show that this problem can be solved in polynomial time. For the proof we construct an auxiliary graph G with directed and undirected edges and show that G has a 4-DORG representation exactly when G has a transitive orientation respecting its directed edges. There is a gap in the proof of Theorem 1. We have not been able to fix itWith an independent approach, we show that if we are given a permutation π of the vertices of U but the partition {L, R} of V \ U is not given, then we can still efficiently check whether G has a 3-DORG representation. Here, π is the order of y-coordinates of endpoints of rays for U .

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“…The set R(G) = {R w | w ∈ V(G)} is called an orthogonal ray representation of G. Orthogonal ray graphs have been introduced in connection with the defect-tolerant design of nano-circuits [9]. An orthogonal ray graph is called a 2-directional orthogonal ray graph (2-DORG for short) if every horizontal ray R u , u ∈ U, has the same direction, and every vertical ray R v , v ∈ V, has the same direction.…”

Section: Introductionmentioning

confidence: 99%

Dominating Sets in Two-Directional Orthogonal Ray Graphs

Takaoka

Tayu

Ueno

2015

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Satoshi TAYU †b) , Members, and Shuichi UENO †c) , Fellow SUMMARY A 2-directional orthogonal ray graph is an intersection graph of rightward rays (half-lines) and downward rays in the plane. We show a dynamic programming algorithm that solves the weighted dominating set problem in O(n 3 ) time for 2-directional orthogonal ray graphs, where n is the number of vertices of a graph.

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Orthogonal ray graphs and nano-PLA design

Cited by 14 publications

References 6 publications

On Two Problems of Nano-PLA Design

On Two Problems of Nano-PLA Design

On the Recognition of Four-Directional Orthogonal Ray Graphs

Dominating Sets in Two-Directional Orthogonal Ray Graphs

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