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.

Abstract. We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs. Key words. common interval, permutation, PQ-tree, modular decomposition AMS subject classifications. 05C05, 05C62, 68R99

Abstract. In 2000, T. Uno and M. Yagiura published an algorithm that computes all the K common intervals of two given permutations of length n in O(n + K) time. Our paper first presents a decomposition approach to obtain a compact encoding for common intervals of d permutations. Then, we revisit T. Uno and M. Yagiura’s algorithm to yield a linear time algorithm for finding this encoding. Besides, we adapt the algorithm to obtain a linear time modular decomposition of an undirected graph, and thereby propose a formal invariant-based proof for all these algorithms. 1

This report surveys the literature on decomposition of binary, multiple-valued, and fuzzy functions. It gives also references to relevant basic logic synthesis papers that concern topics important for decomposition, such as for instance representation of Boolean functions, or symmetry of Boolean functions. As a result of the analysis of the most successful decomposition programs for Ashenhurst-Curtis Decomposition, several conclusions are derived that should allow to create a new program that will be able to outperform all the existing approaches to decomposition. Creating such asuperior program is necessary to make it practically useful for applications that are of interest to Pattern Theory group at Avionics Labs of Wright Laboratories. In addition, the program will be also able to solve problems that have been never formulated before. It will be a test-bed to develop and compare several known and new partial ideas related to decomposition. Our emphasis is on the following topics: 1. representation of data and e cient algorithms for data manipulation, 2. variable ordering methods for variable partitioning to create bound and free sets of input variables � heuristic approaches and their comparison, 3. column compatibility problem, 4. subfunction encoding problem, 5. use of partial and total symmetries in data to decrease the decomposition search space, 6. methods of dealing with strongly unspeci ed functions which are typical for machine learning applications, 7. special types of decomposition, that can be e ciently handled (cascades, trees without variable repetition). 2 We would like toacknowledge Dr. Tim Ross, Mr. Mark Axtell, and Professors Robert Brayton,

Abstract. In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are still efficient. This theory not only unifies the usual modular decomposition generalisations such as modular decomposition of directed graphs and of 2-structures, but also decomposition by star cutsets. 1

We introduces the umodules, a generalisation of the notion of graph module. The theory we develop captures among others undirected graphs, tournaments, digraphs, and 2−structures. We show that, under some axioms, a unique decomposition tree exists for umodules. Polynomial-time algorithms are provided for: non-trivial umodule test, maximal umodule computation, and decomposition tree computation when the tree exists. Our results unify many known decomposition like modular and bi-join decomposition of graphs, and a new decomposition of tournaments. 1

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Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, linear-time algorithm for the problem. Despite considerable effort, such an algorithm has remained elusive. The linear-time algorithms to date are impractical and of mainly theoretical interest. In this paper we present the first simple, linear-time algorithm to compute the modular decomposition tree of an undirected graph. 1

Abstract. We revisit the problem of designing a linear time algorithm for undirected graph split decomposition. Although that this problem has already been claimed to be solved in [E. Dahlhaus, FSTTCS, 1994] and [E. Dahlhaus, Journal of Algorithms 36(2):205-240, 2000], we present a new well founded theoretical background for split decomposition that allow us to clearly design and proove the rst simple linear time split decomposition algorithm. 1

We revisit the problem of designing a linear time algorithm for undirected graph split decomposition that has been rst addressed in [E. Dahlhaus, FSTTCS, 1994] and [E. Dahlhaus, Journal of Algorithms 36(2):205-240, 2000]. We present a new and well founded theoretical background for split decomposition (also known as 1-join decomposition) which allows us to clearly design and prove a relatively simple linear-time split decomposition algorithm. 1