We study bipartite geometric intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we look at various classes of graphs between grid intersection graph and bipartite permutations graphs and the containment relation on these cla...

The Densest k-Subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The DkS problem is NP-hard even for special graph classes including bipartite, planar, comparability and chordal graphs, while no constant approximation algorithm is known for any of these classes. In this paper we present a 3-approximation algorithm for the class of chordal grap...

We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordere...

We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a speci3ed set of processors. Up to three precedence constraints are considered among groups of tasks requiring the same set of processors. The objective of the problem is to 3nd a nonpreemptive schedule which minimizes the maximum completion time (makespan). This sch...

In this paper we introduce the simultaneous membership problem, defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G1 and G2, sharing some vertices X (and the corresponding induced edges), are said to be simultaneous members of graph class C, if there exist representations R1 and R2 of G1 and G2 that are “consistent” on X ....

In the past, diierent authors developed distinct approaches to the problem of transitive orientation. This also resulted in diierent ideas and diierent theorems which seem unrelated. In this paper we show the connections between these theories and present a new algorithm to recognize a comparability graph. A comparability graph is an undirected graph G = (V; E), jV j = n, jEj = m, in which ever...

Given a transitive orientation ~ G of a comparability graph G, a vertex of G is a source (sink) if it has indegree (outdegree) zero in ~ G, respectively. A source set of G is a subset of vertices formed by sources of some transitive orientation of G. A pair of subsets S; T V (G) is a source-sink pair of G when the vertices of S and T are sources and sinks, respectively, of some transitive orien...

Comparability graphs are the undirected graphs whose edges can be directed so that the resulting directed graph is transitive. They are related to posets and have applications in scheduling theory. This paper considers the problem of finding a transitive orientation of a comparability graph, a requirement for many of its applications. A linear-time algorithm is presented based on an elegant par...

Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0, 1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear. We focus on the problem of extending partial representations...

A k-box B = (R1, R2, . . . , Rk), where each Ri is a closed interval on the real line, is defined to be the Cartesian product R1 × R2 × · · · × Rk. If each Ri is a unit length interval, we call B a k-cube. Boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k s...