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 single-source digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a single-source 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.

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 on-line 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.

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 order-theoretic, (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 order-theoretic results. For instance, the transformation P ! Q can be seen as almost an inverse of the well-known 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...

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 vertex-face poset of such a graph.

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).

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