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Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in softwar ..."
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Cited by 34 (4 self)
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Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
New perspectives on interval orders and interval graphs
 in Surveys in Combinatorics
, 1997
"... Abstract. Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the so ..."
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Cited by 7 (5 self)
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Abstract. Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even experimental problems associated with them. In this article, we survey recent work on interval orders and interval graphs, including research on online coloring, dimension estimates, fractional parameters, balancing pairs, hamiltonian paths, ramsey theory, extremal problems and tolerance orders. We provide an outline of the arguments for many of these results, especially those which seem to have a wide range of potential applications. Also, we provide short proofs of some of the more classical results on interval orders and interval graphs. Our goal is to provide fresh insights into the current status of research in this area while suggesting new perspectives and directions for the future. 1.
Adjacency posets of planar graphs
 DISCRETE MATH
"... In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show t ..."
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Cited by 4 (3 self)
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In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertexface poset of such a graph.
Dimension, Graph and Hypergraph Coloring
 ORDER
, 2000
"... There is a natural way to associate with a poset P a hypergraph HP , called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of HP . The ordinary graph GP of incomparable pairs determined by the edges in HP of size 2 can have chromatic number substantially ..."
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Cited by 4 (0 self)
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There is a natural way to associate with a poset P a hypergraph HP , called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of HP . The ordinary graph GP of incomparable pairs determined by the edges in HP of size 2 can have chromatic number substantially less than HP . We give a new proof of the fact that the dimension of P is 2 if and only if GP is bipartite. We also show that for each t 2, there exists a poset P for which the chromatic number of the graph of incomparable pairs is t, but the dimension of P is at least (3=2) t\Gamma1 . However, it is not known whether there is a function f : R ! R so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).
On the Interplay between Interval Dimension and Dimension
, 1991
"... This paper investigates a transformation P ! Q between partial orders P; Q that transforms the interval dimension of P to the dimension of Q, i.e., idim(P ) = dim(Q). Such a construction has been shown before in the context of Ferrer's dimension by Cogis [2]. Our construction can be shown to be equ ..."
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Cited by 3 (2 self)
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This paper investigates a transformation P ! Q between partial orders P; Q that transforms the interval dimension of P to the dimension of Q, i.e., idim(P ) = dim(Q). Such a construction has been shown before in the context of Ferrer's dimension by Cogis [2]. Our construction can be shown to be equivalent to his, but it has the advantage of (1) being purely ordertheoretic, (2) providing a geometric interpretation of interval dimension similar to that of Ore [15] for dimension, and (3) revealing several somewhat surprising connections to other ordertheoretic results. For instance, the transformation P ! Q can be seen as almost an inverse of the wellknown split operation, it provides a theoretical background for the influence of edge subdivision on dimension (e.g., the results of Spinrad [17]) and interval dimension, and it turns out to be invariant with respect to changes of P that do not alter its comparability graph, thus providing also a simple new proof for the comparability in...
THE DIMENSION OF POSETS WITH PLANAR COVER GRAPHS
"... Abstract. Kelly showed that there exist planar posets of arbitrarily large dimension, and Streib and Trotter showed that the dimension of a poset with a planar cover graph is bounded in terms of its height. Here we continue the study of conditions that bound the dimension of posets with planar cover ..."
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Cited by 1 (1 self)
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Abstract. Kelly showed that there exist planar posets of arbitrarily large dimension, and Streib and Trotter showed that the dimension of a poset with a planar cover graph is bounded in terms of its height. Here we continue the study of conditions that bound the dimension of posets with planar cover graphs. We show that if P is poset with a planar comparability graph, then the dimension of P is at most four. We also show that if P has an outerplanar cover graph, then the dimension of P is at most four. Finally, if P has an outerplanar cover graph and the height of P is two, then the dimension of P is at most three. These three inequalities are all best possible. 1.
5 Summary of Results and Open Problems Chapter 2 In Chapter 2, for an infinite graph
"... we defined ..."
Dimensions of Split Semiorders
, 1998
"... A poset P = (X,P) is a split semiorder when there exists a function I that assigns to each x ∈ X aclosedintervalI(x) =[ax,ax + 1] of the real line R and a set F ={fx: x ∈ X} of real numbers, with ax ≤ fx ≤ ax + 1, such that x
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A poset P = (X,P) is a split semiorder when there exists a function I that assigns to each x ∈ X aclosedintervalI(x) =[ax,ax + 1] of the real line R and a set F ={fx: x ∈ X} of real numbers, with ax ≤ fx ≤ ax + 1, such that x<yin P if and only if fx <ay and ax + 1 <fy in R. Every semiorder is a split semiorder, and there are split semiorders which are not interval orders. It is well known that the dimension of a semiorder is at most 3. We prove that the dimension of a split semiorder is at most 6. We note also that some split semiorders have semiorder dimension at least 3, and that every split semiorder has interval dimension at most 2.
An Optimal Ancestry Scheme and Small Universal Posets ∗
"... In this paper, we solve the ancestry problem, which was introduced more than twenty years ago by Kannan et al. [STOC ’88], and is among the most wellstudied problems in the field of informative labeling schemes. Specifically, we construct an ancestry labeling scheme for nnode trees with label size ..."
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In this paper, we solve the ancestry problem, which was introduced more than twenty years ago by Kannan et al. [STOC ’88], and is among the most wellstudied problems in the field of informative labeling schemes. Specifically, we construct an ancestry labeling scheme for nnode trees with label size log 2 n + O(log log n) bits, thus matching the log 2 n + Ω(log log n) bits lower bound given by Alstrup et al. [SODA ’03]. Besides its optimal label size, our scheme assigns the labels in linear time, and guarantees that any ancestry query can be answered in constant time. In addition to its potential impact in terms of improving the performances of XML search engines, our ancestry scheme is also useful in the context of partially ordered sets. Specifically, for any fixed integer k, our scheme enables the construction of a universal poset of size O(n k log 4k n) for the family of nelement posets with treedimension at most k. This bound is almost tight thanks to a lower bound of n k−o(1) due to Alon and Scheinerman [Order ’88].