Partial orders of dimension 2

Baker, Kirby A.; Fishburn, Peter C.; Roberts, Fred S.

doi:10.1002/net.3230020103

Cited by 158 publications

(59 citation statements)

References 15 publications

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“…Using the result by Baker et al [3] we know that such a poset has dimension at most 2. Thus, there are two numberings p and q of the vertices of E * such that a face f is to the left of a face g if and only if p( f ) < p(g) and q( f ) < q(g).…”

Section: Faster Algorithmmentioning

confidence: 92%

The Partial Visibility Representation Extension Problem

et al. 2017

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(u, v) of G. We study a generalization of the recognition problem where a function ψ defined on a subset V of V (G) is given and the question is whether there is a bar visibility representation ψ of G with ψ(v) = ψ (v) for every v ∈ V . We show that for undirected graphs this problem, and other closely related problems, is NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

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Section: Faster Algorithmmentioning

confidence: 92%

The Partial Visibility Representation Extension Problem

et al. 2017

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“…We use some kind of adjacency matrix to show upper bounds of s G,λ 1 (P2) The right and left boundary of a 1-polygon consists of two parts: one is non-decreasing from top to bottom and the other is non-increasing.…”

Section: Biconvex Graphs and Bipartite Permutation Graphsmentioning

confidence: 99%

“…We can assume without loss of generality that the x-coordinates are distinct and the ycoordinates are distinct [1].…”

Section: Permutation 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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“…More generally, any DAG is upward planar if and only if it is a subgraph of a planar st-graph [7]. In the other direction, every planar finite bounded poset must be a lattice [3,5,23]. This implies that a two-dimensional bounded poset that is not a lattice (such as the one on the left of Figure 2) cannot have an upward planar drawing, and that planarity (a crossing-free drawing) and two-dimensionality (realization by a pair of linear orders) are distinct for non-lattice posets.…”

Section: Hasse Diagrams and Upward Planaritymentioning

confidence: 99%

Confluent Hasse Diagrams

Eppstein¹,

Simons²

2013

JGAA

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We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n 2 ) features, in an O(n) × O(n) grid in O(n 2 ) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) features in an O(n) × O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use.

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Partial orders of dimension 2

Cited by 158 publications

References 15 publications

The Partial Visibility Representation Extension Problem

The Partial Visibility Representation Extension Problem

OBDD Representation of Intersection Graphs

Confluent Hasse Diagrams

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