A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.

A module of an undirected graph G = (V, E) is a set X of vertices that have the same set of neighbors in V \ X. The modular decomposition is a unique decomposition of the vertices into nested modules. We give a practical algorithm with an O(n + m(m;n)) time bound and a variant with a linear time bound.

When a parallel computation is represented in a formalism that imposes series-parallel structure on its task graph, it becomes amenable to automated analysis and scheduling. Unfortunately, its execution time will usually also increase as precedence constraints are added to ensure series-parallel structure. Bounding the slowdown ratio would allow an informed tradeoff between the benefits of a restrictive formalism and its cost in loss of performance. This dissertation deals with series-parallelising task graphs by adding precedence constraints to a task graph, to make the resulting task graph series-parallel. The weak bounded slowdown conjecture for series-parallelising task graphs is introduced. This states that the slowdown is bounded if information about the workload can be used to guide the selection of which precedence constraints to add. A theory of best series-parallelisations is developed to investigate this conjecture. Partial evidence is presented that the weak slowdown bound is likely to be 4/3, and this bound is shown to be tight.

Two edges are called P 4 -adjacent if they belong to the same P 4 (chordless path on 4 vertices). P 4 -components, in our terminology, are the equivalence classes of the transitive closure of the P 4 -adjacency relation. In this paper, new results on the structure of P 4 -components are obtained. On the one hand, these results allow us to improve the complexity of the recognition and orientation algorithms for P 4 -comparability and P 4 - indifference graphs from O(n 5 ) to O(n 2 m) and from O(n 6 ) to O(n 2 m), respectively. On the other hand, by combining the modular decomposition with the substitution of P 4 - components, a new unique tree representation for arbitrary graphs is derived which generalizes the homogeneous decomposition introduced by Jamison and Olariu [JO95]. 1 Introduction A P k (C k ) is a chordless path (cycle) on k vertices. By the P 4 abcd, we denote the P 4 with vertices a; b; c; d and edges ab, bc and cd. An orientation U of a graph G is the antisymmet...

We consider graph applications of the well-known paradigm "killing two birds with one stone". In the plane, this gives rise to a stage graph as follows: vertices are the points, and fu; vg is an edge if and only if the (infinite, straight) line segment joining u to v intersects the stage. Such graphs are shown to be comparability graphs of ordered sets of dimension 2. Similar graphs can be constructed when we have a fixed number k of stages on the plane. In this case, fu; vg is an edge if and only if the (straight) line segment uv intersects one of the k stages. We study stage representations of stage graphs and give upper and lower bounds on the number of stages needed to represent a graph. 1980 Mathematics Subject Classification: 68R10, 68U05 CR Categories: F.2.2 Key Words and Phrases: Algorithms, Girth, Ray shooting, Partial Orders, Stage Graphs. Carleton University, School of Computer Science: SCS-TR-95-07 Note: This technical report is a revised and expanded version of TR-23...

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 orientation of G. We describe algorithms for computing the cardinality of a maximum source set and of a maximum source-sink pair of a comparability graph. In addition, we describe algorithms for finding the corresponding transitive orientations. The algorithms are applications of modular decomposition and are all of linear-time complexity. Keywords: algorithms, comparability graphs, modular decomposition, source sets, transitive orientation. 1 Introduction Let G denote a simple non-trivial connected undirected graph, with vertex set V (G) and edge set E(G). Write n = jV (G)j and m = jE(G)j. L...

This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a well-known optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear equalities and inequalities. The state of the art in solvers for this problem is the “branch and bound ” approach. The performance of such solvers depends crucially on four types of in-built heuristics: primal, improvement, branching, and cut-separation or, more generally, bounding heuristics. Such heuristics in general-purpose solvers have not, until recently, exploited structure in integer linear programs beyond the recognition of certain types of single-row constraints. Many alternative approaches to integer linear programming can be cast in the following, novel framework. “Structure” in any integer linear program