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.

We show that for every interval order X, there exists an integer t so that if Y is any interval order with dimension at least t, then Y contains a subposet isomorphic to X. c ○ 2000 Published

Characterizing posets for which their natural

Characterizing posets for which their natural transit functions coincide*

Abstract not found

In this paper we study the 2-dimension of a finite poset from the topological point of view. We use homotopy theory of finite topological spaces and the concept of a beat point to improve the classical results on 2-dimension, giving a more complete answer to the problem of all possible 2-dimensions of an n-point poset.

Abstract. The threshold dimension of a graph G(V, E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A k-dimensional box is the Cartesian product R1 × R2 × · · · × Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R1 × R2 × · · · × Rk where each Ri is a closed interval on the real line of the form [ai, ai + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. In this paper we will show that there exists no polynomial-time algorithm to approximate the threshold dimension of a graph on n vertices with a factor of O(n 0.5−ǫ) for any ǫ> 0, unless NP = ZPP. From this result we will show that there exists no polynomial-time algorithm to approximate the boxicity and the cubicity of a graph on n vertices with factor O(n 0.5−ǫ) for any ǫ> 0, unless NP = ZPP. In fact all these hardness results hold even for a highly structured class of graphs namely the split graphs. We will also show that it is NP-complete to determine if a given split graph has boxicity at most 3.