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.

We extend the multiplicative fragment of linear logic with a non-commutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae. We firstly examine coherence semantics, where we introduce the before connective, and ordered products of formulae. Secondly we extend the syntax of multiplicative proof nets to these new operations. We then prove strong normalisation, and confluence. Coming back to the denotational semantics that we started with, we establish in an unusual way the soundness of this calculus with respect to the semantics. The converse, i.e. a kind of completeness result, is simply stated: we refer to a report for its lengthy proof. We conclude by mentioning more results, including a sequent calculus which is interpreted by both the semantics and the proof net syntax, although we are not sure that it takes all proof nets into account...

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.

Lambek calculus may be viewed as a fragment of linear logic, namely intuitionistic non-commutative multiplicative linear logic. As it is too restrictive to describe numerous usual linguistic phenomena, instead of extending it we extend MLL with a non-commutative connective, thus dealing with partially ordered multisets of formulae. Relying on proof net technique, our study associates words with parts of proofs, modules, and parsing is described as proving by plugging modules. Apart from avoiding spurious ambiguities, our method succeeds in obtaining a logical description of relatively free word order, head-wrapping, clitics, and extraposition (these latest two constructions are unfortunately not included, for lack of space).

A dominating set S of a graph G is perfect if each vertex of G is dominated by exactly one vertex in S. We study the existence and construction of PDSs in families of graphs arising from the interconnection networks of parallel computers. These include trees, dags, series-parallel graphs, meshes, tori, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. For trees, dags, and series-parallel graphs we give linear time algorithms that determine if a PDS exists, and generate a PDS when one does. For 2- and 3-dimensional meshes, 2-dimensional tori, hypercubes, and cube-connected paths we completely characterize which graphs have a PDS, and the structure of all PDSs. For higher dimensional meshes and tori, cube-connected cycles, and de Bruijn graphs, we show the existence of a PDS in infinitely many cases, but our characterization is not complete. Our results include distance d-domination for arbitrary d. 1 Introduction Suppose G = (V; E) is a graph with vertex se...

Series-parallel orders are defined as the least class of partial orders containing the one-element order and closed by ordinal sum and disjoint union. From this inductive denition, it is almost immediate that any series-parallel order may be represented by an algebraic expression, which is unique up to the associativivity of ordinal sum and to the associativivity and commutativity of disjoint union. In this paper, we introduce a rewrite system acting on these algebraic expressions that axiomatises completely the sub-ordering relation for the class of series-parallel orders.

Given a stochastic project network with independently distributed activity durations, several approaches to bound the distribution function of the project completion time have been proposed. We have implemented the most promising of these algorithms and compare their behavior on a basis of nearly 2000 instances with up to 1200 activities of different test-beds. We propose a suitable numerical representation of the given distributions which is the basis for excellent computational results.

. This paper considers languages in a free algebra which has a binary associative operation called the series product. We define automata operating in these algebras and rational expressions, and we show that their expressive powers coincide (a Kleene theorem). We also show that this expressive power equals that of algebraic recognizability (a Myhill-Nerode theorem). This generalizes the work of Thatcher and Wright. The first equivalence continues to hold when conditions such as associativity and commutativity are imposed on the term operations, but recognizability is weaker when one of the term operations (other than the series product) is associative. We also consider languages which have a bound on the number of nested occurrences of certain designated term operations and get both the equivalences mentioned above. This generalizes our earlier work and answers a question left open therein. 1 Automata form one of the most commonly used computing device, from their histor...

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.

. In order to model concurrency, we extend automata theory from the usual word languages (sets of labelled linear orders) to sets of labelled series-parallel posets --- or, equivalently, to sets of terms in an algebra with two product operations: sequential and parallel. We first consider languages of posets having bounded width, and characterize them using depth-nilpotent algebras. Next we introduce series-rational expressions, a natural generalization of the usual rational expressions, as well as a notion of branching automaton. We show both a Myhill-Nerode theorem and a Kleene theorem. We also look at generalizations. Introduction In this paper, we seek to extend automata theory from the usual languages over words (labelled linearly ordered sets) to (labelled) posets which include more information regarding concurrency. This has been done earlier; we explain how our approach differs below. Let A be a finite nonempty alphabet. A language over A accepted by a finite automaton is said...