Loÿs Thimonier

Forbidden subgraphs in connected graphs

Given a set ξ = {H1,H2,...} of connected non-acyclic graphs, a ξ-free graph is one which does not contain any member of ξ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let Wk,ξ be the exponential generating function (EGF for brief) of connected ξ-free graphs of excess equal to k (k ≥ 1). For each fixed ξ, a fundamental differential recurrence satisfied by the EGFs Wk,ξ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of k by means of graph surgery. We also show that for any finite collection ξ of non-acyclic graphs, the EGFs Wk,ξ are always rational functions of the generating function, T, of Cayleys rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with n nodes and n + k edges are ξ-free, whenever k = o(n1/3) and |ξ| < ∞ by means of Wrights inequalities and saddle point method. Limiting distributions are derived for sparse connected ξ-free components that are present when a random graph on n nodes has approximately n/2 edges. In particular, the probability distribution that it consists of trees, unicyclic components, ..., (q + 1)-cyclic components all ξ-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed.

Using Generating Functions to Compute Concurrency

The aim of this work is the improvement of the computation of the concurrency measure defined by Beauquier, Berard and Thimonier in [BT87]. First, we present the Arnold-Nivats model for the synchronization of sequential processes, and we recall the formal definition of the concurrency measure. Then, we present a new technique for the computation of this measure often avoiding a very expensive part of the computation. The mutual exclusion example illustrates this method.

On formal languages, probabilities, paging and decoding algorithms

In previous papers [BT 83] [BT 84] [BT 85], it was indicated how a probabilistic parameter, namely the Bernoullian density, could be computed by means of an explicit formula or numerically with a given precision for several structural types of formal Languages L. We present here two general methods for this computing, a deterministic one and a Monte-Carlo method, using the generating function of the prefix-free language Pref(L) associated with L, and a recognition algorithm for Pref(L) assumed to be context-free. The results are applied for the obtainment of new algorithms for two classical problems : paging and decoding, within a probabilistic framework of language theory.

Computability of Probabilistic Parameters for Some Classes of Formal Languages

In a previous paper [BT 83], some probabilistic notions of density and waiting time for a formal language have been studied. We prove here that these probabilistic parameters are computable with an arbitrary precision for some families of languages : the languages with an end marker ; the prefix-free regular sets, with matricial algorithms on Markov chains related to deterministic finite-state automata ; at the end, the prefix-free languages of palindrom words, for which the use of counting generating series yields new results in the equally likely case, already studied in [BT 83], and allows to give partial answers in the general case.

Asymptotic enumeration of cographs

Abstract Abstract We consider here labelled and unlabelled cographs, i.e., graphs without induced P4, whose applications are important in computer science and logic, because of their representation by means of parse trees. After a new (analytical) approach for obtaining generating functions associated to parse trees, we solve the open problem of asymptotic estimates for the numbers of unlabelled and labelled cographs.

Some Remarks on Sparsely Connected Isomorphism-Free Labeled Graphs

Given a set ξ = {H1,H2, ⋯ } of connected non-acyclic graphs, a ξ-free graph is one which does not contain any member of ξ as induced subgraph. Our first purpose in this paper is to perform an investigation into the limiting distribution of labeled graphs and multigraphs (graphs with possible self-loops and multiple edges), with n vertices and approximately \(\frac{1}{2}n\) edges, in which all sparse connected components are ξ-free. Next, we prove that for any finite collection ξ of multicyclic graphs almost all connected graphs with n vertices and n + o(n1/3) edges are ξ -free. The same result holds for multigraphs.

Breadth First Search, Triangle-Free Graphs and Brownian Motion

One major problem in the enumeration of random graphs concerns triangle-free graphs. In this paper we study Breadth First Search processes and the associated queues to compute in terms of Wright’s constants the number of triangle-free graphs. Next we prove that this number is equivalent to the number connected labelled graphs by using arguments of the Brownian excursion type.

A Common Asymptotic Behavior for Different Classes of Sparse Labelled Graphs with Given Number of Vertices and Edges

Let m(n, n + k) be the number of connected labelled multi-graphs, which are graphs with n vertices, n + k edges and possible self-loops and/or multiple edges. Denote by c(n, n + k) the number of connected labelled simple graphs with the same parameters. First, under the condition that k = o(n 2), by making use of the methods developped by Bender et al. in [3], we show that m(n, n + k) ≈ c(n,n + k) as n → ∞. Under the same condition on the number of exceeding edges, k = o(n 2), these results are extended to show that connected labelled graphs, multi-graphs and graphs without a finite set of forbidden subgraphs have the same asymptotic behavior. Finally, we give sufficient condition, in terms of the total number of graphs, for the probability of connectedness to have a limit equal to 1 as the number of vertices tends to ∞.

A Markovian concurrency measure

We show how to modelize concurrency between several processors in terms of automata and Markov chains; then, we define a concurrency measure which reflects more faithfully the behaviour of the processes and is in addition easy to compute with a symbolic manipulator like Maple (this is an improvement over a previous measure [3] whose computation appears to be expensive.

Fonctions Génératrices Transcendantes à Coefficients Engendrés par Automates

Soit (u(n)) une suite de ±1 engendree par un 2-automate, nous montrons que la fonction generatrice associee