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 graph G is a circular-arc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a linear-time algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.

A 0-1 matrix has the consecutive-ones property if its columns can be ordered so that the ones in every row are consecutive. It has the circular-ones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutive-ones orderings of the columns of a matrix that has the consecutive-ones property. We give an analogous structure, called a PC tree, for representing all circular-ones orderings of the columns of a matrix that has the circular-ones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1

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.

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.

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.

4> : : : : : : : : : : : : : : : : : : : : : : : 23 1.5 Partial orderings : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 1.6 Lattices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 1.7 Equivalence relations and partitions : : : : : : : : : : : : : : : : : 24 1.8 Lattices as algebras : : : : : : : : : : : : : : : : : : : : : : : : : : 25 1.8.1 Lattice varieties : : : : : : : : : : : : : : : : : : : : : : : : 25 II Morphisms of graphs 27 2.1 Graph morphisms : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 2.2 An induction principle : : : : : : : : : : : : : : : : : : : : : : : : 32 2.3 Modules and morphisms : : : : : : : : : : : : : : : : : : : : : : : 33 2.4 Comparison with strong homomorphisms : : : : : : : : : : : : : : 34 2.5 Strongly connected graphs : : : : : : : : : : : : : : : :

We propose an O(n ) algorithm to build the modular decomposition tree of hypergraphs of dimension 3 and show how this algorithm can be generalized to compute in O(n ) time the decomposition of hypergraphs of any fixed dimension k.