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.

An interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each pair of intersecting intervals. A probe interval graph is obtained from an interval graph by designating a subset P of vertices as probes, and removing the edges between pairs of vertices in the remaining set N of non-probes. We examine the problem of finding and representing possible layouts of the intervals, given a probe interval graph. We obtain an O(n + m log n) bound, where n is the number of vertices and m is the number of edges. The problem is motivated by an application to molecular biology.

The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NP-complete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement of a bipartite graph. The problem can be solved in polynomial time, when restricted to graphs with bounded treewidth, or cographs. We also give large classes of graphs that can be seen as generalizations of classes of graphs with bounded treewidth and of the class of the cographs, and allow polynomial time algorithms for the simple max cut problem. 1 Introduction One of the best known combinatorial graph problems is the max cut problem. In this problem, we have a weighted, undirected graph G = (V; E) and we look for a partition of the vertices of G into two disjoint sets, such that the total weight of the edges that go from one set to the other is as large as possible. In the simple max cu...

Abstract. In this paper, we present a linear-time algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a linear-time algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O(n 2)toO(n+m). We also devise a simple linear-time algorithm for interval graph recognition where no complicated data structure is involved. Key words. chordal graph, triangulated graph, interval graph, analysis of algorithms, graph theory, substitution decomposition, modular decomposition, cycle-free poset, transitive orientation, graph partitioning, cardinality lexicographic ordering, graph recognition

This paper gives an O(n²) incremental algorithm for computing the modular decomposition of 2-structure [1, 2]. A 2-structure is a type of edge-colored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2-structures arises in the study of relational systems. The modular decomposition of undirected graphs and digraphs is a special case, and has applications in a number of combinatorial optimization problems. This algorithm generalizes elements of a previous O(n²) algorithm of Muller and Spinrad [3] for the decomposition of undirected graphs. However, Muller and Spinrad's algorithm employs a sophisticated data structure that impedes its generalization to digraphs and 2-structures, and limits its practical use. We replace this data structure with a scheme that labels each edge with at most one node, thereby obtaining an algorithm that is both practical and general to 2-structures.

The graph sandwich problem for property \Phi is defined as follows: Given two graphs G 1 = (V; E 1 ) and G 2 = (V; E 2 ) such that E 1 ` E 2 , is there a graph G = (V; E) such that E 1 ` E ` E 2 which satisfies property \Phi? We present a polynomial-time algorithm for solving the graph sandwich problem, when property \Phi is "to contain a homogeneous set". The algorithm presented also provides the graph G and a homogeneous set in G in case it exists. Keywords: design of algorithms, sandwich problems, perfect graphs. 1 Introduction We say that a graph G 1 = (V; E 1 ) is a spanning subgraph of G 2 = (V; E 2 ) when E 1 ` E 2 ; we say that a graph G = (V; E) is a sandwich graph for the pair G 1 , G 2 when E 1 ` E ` E 2 . According to [6], the graph sandwich problem for property \Phi is defined as follows: Problem: graph sandwich problem for property \Phi Instance: Two graphs, G 1 and G 2 , such that G 1 is a spanning subgraph of G 2 . Question: Does there exist a sandwich graph for the...

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.

this paper, we explore algorithmic uses of this concept on graphs and digraphs. An outward complementation operation is where only the outgoing arcs of a vertex are complemented. That is, the neighbors of the vertex are turned into non-neighbors and the non-neighbors are turned into neighbors. An inward complementation operations is where only the inward arcs are complemented. A symmetric complementation operation is one where both the inward and the outward arcs are complemented.

Abstract not found