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.

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.

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This paper deals with the differences between the modular and the homogeneous decomposition of graphs. It is shown how the homogeneous decomposition can be derived from the modular decomposition. As this method works in linear time the available linear algorithms for the modular decomposition are extended to a linear algorithm for the homogeneous decomposition of arbitrary graphs. Therefore, this paper generalizes several results on the decomposition of special graph-classes like cographs, P 4 -sparse graphs, P 4 -reducible graphs or P 4 -extendible graphs.

Abstract — This paper presents a heuristic algorithm for disjoint decomposition of a Boolean function based on its ROBDD representation. Two distinct features make the algorithm feasible for large functions. First, for an n-variable function, it checks only O(n 2) candidates for decomposition out of O(2 n) possible ones. A special strategy for selecting candidates makes it likely that all other decompositions are encoded in the selected ones. Second, the decompositions for the approved candidates are computed using a novel IntervalCut algorithm. This algorithm does not require re-ordering of ROBDD. The combination of both techniques allows us to decompose the functions of size beyond that possible with the exact algorithms. The experimental results on 582 benchmark functions show that the presented heuristic finds 95 % of all decompositions on average. For 526 of those functions, it finds 100 % of the decompositions. I.

. In 1985, Corneil, Perl and Stewart [CPS85] gave a linear incremental algorithm to recognize cographs (graphs with no induced P4 ). When this algorithm stops, either the initial graph is a cograph and the cotree of the whole graph has been built, or the initial graph is not a cograph and this algorithm ends up with a vertex v and a cotree cot such that v cannot be inserted in cot; so the input graph must contain a P4 . In many applications such as graph decomposition [Cou93, CH93a, CH93b, CH94, EGMS94, Spi92, MS94], transitive orientation [Spi83, ST94], not only the existence but a P4 is also explicitly needed. In this paper, we present a new characterization of cograph in terms of its modular structure. This characterization yields a structural labeling of the cotree for incremental cograph recognition, and we show how to go from this labeling to the Corneil et al. one's. Furthermore, we show how to adapt this algorithm in order to produce a P4 in case of failure when adding a new v...

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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

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