Publications (115)29.98 Total impact

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ABSTRACT: We consider the class of graphs having linear rankwidth one, also known as thread graphs, and investigate a related graph modification problem called the Thread Vertex Deletion. In this problem, given an $n$ vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a thread graph and if one exists, find such a vertex set. While the metatheorem of Courcelle, Makowsky, Rotics implies that Thread Vertex Deletion can be computed in time $f(k)\cdot n^3$, it is not clear whether this problem allows a runtime with a modest exponential function. We establish that Thread Vertex Deletion can be solved in time $8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define a graph class called the necklace graphs and investigate its structural properties. We also show that the Thread Vertex Deletion has a polynomial kernel. 
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ABSTRACT: The Hadwiger number of a graph G is the largest integer h such that G has the complete graph K_h as a minor. We show that the problem of determining the Hadwiger number of a graph is NPhard on cobipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most $s$. We show that this problem can be solved in polynomial time on ATfree graphs when s>=2, but is NPhard on chordal graphs for every fixed s>=2. 
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ABSTRACT: The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open. 
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ABSTRACT: Several algorithmic metatheorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive metakernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for rDominating Set and rScattered Set on apexminorfree graphs, and for PlanarFDeletion on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected. 
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ABSTRACT: Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of levelk networks. In particular, we give an equivalence theorem between circular split systems and unrooted level1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted levelk phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.Journal of Bioinformatics and Computational Biology 08/2012; 10(4):1250004. DOI:10.1142/S0219720012500047 · 0.93 Impact Factor 
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ABSTRACT: A \emph{$t$treewidthmodulator} of a graph $G$ is a set $X \subseteq V(G)$ such that the treewidth of $GX$ is at most some constant $t1$. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs $G$ that come equipped with a $t$treewidthmodulator. This decomposition, called a \emph{protrusion decomposition}, is the cornerstone in obtaining the following two main results. We first show that any parameterized graph problem (with parameter $k$) that has \emph{finite integer index} and is \emph{treewidthbounding} admits a linear kernel on $H$topologicalminorfree graphs, where $H$ is some arbitrary but fixed graph. A parameterized graph problem is called treewidthbounding if all positive instances have a $t$treewidthmodulator of size $O(k)$, for some constant $t$. This result partially extends previous metatheorems on the existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS 2009] and $H$minorfree graphs [Fomin et al., SODA 2010]. Our second application concerns the Planar$\mathcal{F}$Deletion problem. Let $\mathcal{F}$ be a fixed finite family of graphs containing at least one planar graph. Given an $n$vertex graph $G$ and a nonnegative integer $k$, Planar$\mathcal{F}$Deletion asks whether $G$ has a set $X\subseteq V(G)$ such that $X\leq k$ and $GX$ is $H$minorfree for every $H\in \mathcal{F}$. Very recently, an algorithm for Planar$\mathcal{F}$Deletion with running time $2^{O(k)} n \log^2 n$ (such an algorithm is called \emph{singleexponential}) has been presented in [Fomin et al., FOCS 2012] under the condition that every graph in $\mathcal{F}$ is connected. Using our algorithm to construct protrusion decompositions as a building block, we get rid of this connectivity constraint and present an algorithm for the general Planar$\mathcal{F}$Deletion problem running in time $2^{O(k)} n^2$. 
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ABSTRACT: A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):5163, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NPcomplete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction:  Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]hard in circle graphs, parameterized by the size of the solution.  Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomialtime solvable in circle graphs.  If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NPcomplete when T is in the input, and FPT when parameterized by V(T). We prove that the FPT algorithm is subexponential.Theory of Computing Systems 05/2012; DOI:10.1007/s0022401394788 · 0.45 Impact Factor 
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ABSTRACT: Given an input graph G and an integer k, the parameterized K_4minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K_4minorfree graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two wellstudied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidtht Vertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set. While a singleexponential FPT algorithm has been known for a long time for \textsc{Vertex Cover}, such an algorithm for Feedback Vertex Set was devised comparatively recently. While it is known to be unlikely that Treewidtht Vertex Deletion can be solved in time c^{o(k)}.n^{O(1)}, it was open whether the K_4minor cover problem could be solved in singleexponential FPT time, i.e. in c^k.n^{O(1)} time. This paper answers this question in the affirmative.Journal of Computer and System Sciences 04/2012; 81(1). DOI:10.1007/9783642311550_11 · 1.09 Impact Factor 
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ABSTRACT: Some of the most well studied problems in algorithmic graph theory deal with modifying a graph into an acyclic graph or into a path, using as few operations as possible. In Feedback Vertex Set and Longest Induced Path, the only allowed operation is vertex deletion, and in Spanning Tree and Longest Path, only edge deletions are permitted. We study the edge contraction variant of these problems: given a graph G and an integer k, decide whether G can be transformed into an acyclic graph or into a path using at most k edge contractions. Both problems are known to be NPcomplete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. On the negative side, both problems remain NPcomplete when restricted to bipartite input graphs.Discrete Applied Mathematics 08/2011; 37:8792. DOI:10.1016/j.endm.2011.05.016 · 0.68 Impact Factor 
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ABSTRACT: Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. Interestingly, the study of edge contraction problems of this type from a parameterized perspective has so far been left largely unexplored. We consider two basic edge contraction problems, which we call PathContractibility and TreeContractibility. Both problems take an undirected graph G and an integer k as input, and the task is to determine whether we can obtain a path or an acyclic graph, respectively, by contracting at most k edges of G. Our main contribution is an algorithm with running time 4 k+O(log 2 k) + n O(1) for PathContractibility and an algorithm with running time 4.88 k n O(1) for TreeContractibility, based on a novel application of the color coding technique of Alon, Yuster and Zwick. Furthermore, we show that PathContractibility has a kernel with at most 5k + 3 vertices, while TreeContractibility does not have a polynomial kernel unless coNP ⊆ NP/poly. We find the latter result surprising, because of the strong connection between TreeContractibility and Feedback Vertex Set, which is known to have a vertex kernel with size O(k 2).Algorithmica 05/2011; 68(1). DOI:10.1007/9783642280504_5 · 0.57 Impact Factor 
Article: Hitting and Harvesting Pumpkins
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ABSTRACT: The "cpumpkin" is the graph with two vertices linked by c>0 parallel edges. A cpumpkinmodel in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing cpumpkinmodels in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all cpumpkinmodels can be covered by at most k vertices. This generalizes known singleexponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)approximation algorithm for both the problems of covering all cpumpkinmodels with a smallest number of vertices, and packing a maximum number of vertexdisjoint cpumpkinmodels.SIAM Journal on Discrete Mathematics 05/2011; 28(3). DOI:10.1137/120883736 · 0.58 Impact Factor 
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ABSTRACT: Circle graphs are the intersection graphs of chords in a circle. This paper presents the first subquadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n + m) times the inverse Ackermann function, {\alpha}(n + m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic BreadthFirst Search characterization of circle graphs, and a new efficient datastructure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the authors in a companion paper.Algorithmica 04/2011; 69(4). DOI:10.1007/s0045301397458 · 0.57 Impact Factor 
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ABSTRACT: Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graphlabelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic BreadthFirst Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)\alpha(n+m)$, where $\alpha$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' lineartime split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first subquadratic circle graph recognition algorithm.Algorithmica 04/2011; DOI:10.1007/s0045301397529 · 0.57 Impact Factor 
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ABSTRACT: We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. Our main result is an $f(k) n^{O(1)}$ time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. Our algorithm is based on a novel combination of the irrelevant vertex technique, introduced by Robertson and Seymour, and the concept of important separators. Both techniques have previously been used as key components of algorithms for fundamental problems in parameterized complexity. However, to the best of our knowledge, this is the first time the two techniques are applied in unison.SIAM Journal on Discrete Mathematics 02/2011; DOI:10.4230/LIPIcs.FSTTCS.2011.217 · 0.58 Impact Factor 
Article: Conflict Packing yields linear vertexkernels for Rooted Triplet Inconsistency and other problems
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ABSTRACT: We develop a technique that we call Conflict Packing in the context of kernelization. We illustrate this technique on several wellstudied problems: Feedback Arc Set in Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T leads to an acyclic tournament. While a linear vertexkernel is already known for this problem [6], using the Conflict Packing allows us to find a socalled safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the Dense Rooted Triplet Inconsistency problem, one is given a set of vertices V and a dense collection R of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R. Using again the Conflict Packing, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertexkernel. This result improves the best known bound of O(k^2) vertices for this problem [19]. Finally, we use this technique to obtain a linear vertexkernel for Betweenness in Tournaments, where one is given a set of vertices V and a dense collection R of betweenness triplets and seeks an ordering containing all but at most k triplets from R. To the best of our knowledge this result constitutes the rst polynomial kernel for the problem. 
Conference Paper: Conflict Packing Yields Linear VertexKernels for k FAST, k dense RTI and a Related Problem.
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ABSTRACT: We develop a technique that we call Conflict Packing in the context of kernelization [7]. We illustrate this technique on several wellstudied problems: Feedback Arc Set in Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T results in an acyclic tournament. While a linear vertexkernel is already known for this problem [6], using the Conflict Packing allows us to find a socalled safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the Dense Rooted Triplet Inconsistency problem, one is given a set of vertices V and a dense collection R\mathcal{R} of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R\mathcal{R}. Using again the Conflict Packing, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertexkernel. This result improves the best known bound of O(k 2) vertices for this problem [16]. Finally, we use this technique to obtain a linear vertexkernel for Betweenness in Tournaments, where one is given a set of vertices V and a dense collection R\mathcal{R} of betweenness triplets and seeks an ordering containing all but at most k triplets from R\mathcal{R}. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.Mathematical Foundations of Computer Science 2011  36th International Symposium, MFCS 2011, Warsaw, Poland, August 2226, 2011. Proceedings; 01/2011 
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ABSTRACT: A graph G=(V,E) is a 3leaf power iff there exists a tree T the leaf set of which is V and such that uv∈E iff u and v are at distance at most 3 in T. The 3leaf power graph edge modification problems, i.e. edition (also known as the closest 3leaf power), completion and edgedeletion are FPT when parameterized by the size of the edge set modification. However, polynomial kernels were known for none of these three problems. For each of them, we provide kernels with O(k3) vertices that can be computed in linear time. We thereby answer an open problem first mentioned by Dom et al. (2004) [8].Discrete Applied Mathematics 08/2010; 158(16158):17321744. DOI:10.1016/j.dam.2010.07.002 · 0.68 Impact Factor 
Conference Paper: On the (Non)existence of Polynomial Kernels for
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ABSTRACT: Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P{\it \Pi} consists in deciding whether there exists a set of edges F of size at most k such that the graph H=(V,E\vartriangle F)H=(V,E\vartriangle F) satisfies the property P{\it \Pi}. In the P{\it \Pi} edgecompletion problem, the set F of edges is constrained to be disjoint from E; in the P{\it \Pi} edgedeletion problem, F is a subset of E; no constraint is imposed on F in the P{\it \Pi} edgeediting problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P{\it \Pi} is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P{\it \Pi} edgemodification problems are FPT [4]. It was then natural to ask [4] whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively [15]. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths and pointed out that the problem is already open in the case of P 4free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for P l free and C l free edge deletion problems for large enough l.Parameterized and Exact Computation  5th International Symposium, IPEC 2010, Chennai, India, December 1315, 2010. Proceedings; 07/2010 
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ABSTRACT: Cluster Editing is a classical graph theoretic approach to tackle the problem of data set clustering: it consists of modifying a similarity graph into a disjoint union of cliques, i.e, clusters. As pointed out in a number of recent papers, the cluster editing model is too rigid to capture common features of real data sets. Several generalizations have thereby been proposed. In this paper, we introduce (p, q)cluster graphs, where each cluster misses at most p edges to be a clique, and there are at most q edges between a cluster and other clusters. Our generalization is the ﬁrst one that allows a large number of false positives and negatives in total, while bounding the number of these locally for each cluster by p and q. We show that recognizing (p, q)cluster graphs is NPcomplete when p and q are input. On the positive side, we show that (0, q)cluster, (p, 1)cluster, (p, 2)cluster, and (1, 3)cluster graphs can be recognized in polynomial time. 
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ABSTRACT: Modular decomposition is a technique that applies to (but is not restricted to) graphs. The notion of a module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 1970’s, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research.Computer Science Review 02/2010; DOI:10.1016/j.cosrev.2010.01.001
Publication Stats
1k  Citations  
29.98  Total Impact Points  
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Institutions

2006–2012

French National Centre for Scientific Research
Lutetia Parisorum, ÎledeFrance, France


2003–2012

Université Montpellier 2 Sciences et Techniques
Montpelhièr, LanguedocRoussillon, France


1999–2011

Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM)
Montpelhièr, LanguedocRoussillon, France


2009

University of Tuebingen
 Center for Bioinformatics
Tübingen, BadenWürttemberg, Germany


2007

McGill University
 School of Computer Science
Montréal, Quebec, Canada


2000

National Institute for Research in Computer Science and Control
Le Chesney, ÎledeFrance, France
