*Organisation.*

Fabio Furini et Florian Sikora

*Liste de diffusion.*

Possibilité de s’inscrire à la liste de diffusion en envoyant un email à :

florian . sikora @ dauphine . fr

Les séminaires ont lieu en général le lundi à 14h.

*Prochains séminaires :*

### lundi 11 décembre 2017, 14h, A 304 : Ararat Harutyunyan (LAMSADE) : A disproof of the normal graphs conjecture

Abstract : A graph G is called *normal* if there exist two coverings, C and S, of its vertex set such that every member of C induces a clique in G, every member of S induces an independent set in G, and any clique in C and independent set in S have a non-empty intersection. Normal graphs derive their motivation from information theory, where they are related to the Shannon capacity of a graph. In particular, they form a family which extends the class of perfect graphs. It was conjectured by De Simone and Körner [DAM ’99] that a graph G is normal if G does not contain C_5, C_7 and the complement of C_7 as an induced subgraph. Using random graphs and rather routine probabilistic methods, we give a disproof of this conjecture.

Joint work with Lucas Pastor (G-SCOP, Grenoble) and Stéphan Thomassé (ENS Lyon).

### date à préciser : Benoît Gaüzère (LITIS) : Graph edit distance as a quadratic assignment problem

(séminaire commun avec le pôle 3)

Graphs allow to encode structural information included within

data used in chemical or pattern recognition problems. However,

conversely to vectors defined in an euclidean space, the

definition of a graph (dis)similarity measure is not

straightforward, but required to compute prediction models. One

of the most well known dissimilarity measure is the graph edit

distance. Despite its good interpretability, the computation of a

graph edit distance between two graphs is an NP-Hard

problem. Therefore, its application remains limited to small

graphs. During this presentation, I will introduce a formal

definition of this metric between graphs as a quadratic

assignment problem and some methods used in pattern recognition

to approximate an optimal solution. Considering approximations

allows us to apply this framework to chemoinformatics problems.

*Séminaires précédents :*

### lundi 20 novembre 2017, 14h, P 301 : Julien Baste (LIP6) : F-M-DELETION parameterized by treewidth

For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a set S of vertices of G of size at most k such that G without the vertices of S does not contain any of the graphs of F as a minor. This problem is a generalization of some well known problems as VERTEX COVER (F=*K_2*), FEEDBACK VERTEX SET (F=*K_3*), or VERTEX PLANARIZATION (F=*K_5, K_ 3,3 *). We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of the input graph, denoted by tw. Our objective is to determine, for a fixed F, the smallest function f such that F-M-DELETION can be solved in time f(tw)*poly(n) on n-vertex graphs.

### lundi 13 novembre 2017, 14h, salle A (C206, 2ème étage) : Tomáš Toufar (Charles University, Czech Republic) : Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

We study the Steiner Tree problem, in which a set of terminal vertices

needs to be connected in the cheapest possible way in an edge-weighted

graph. This problem has been extensively studied from the viewpoint of

approximation and also parametrization. In particular, on one hand

Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if

parameterized by the number of non-terminals (Steiner vertices) in the

optimum solution. In contrast to this we

give an efficient parameterized approximation scheme (EPAS), which

circumvents both hardness results. Moreover, our methods imply the

existence of a polynomial size approximate kernelization scheme

(PSAKS) for the assumed parameter.

We further study the parameterized approximability of other variants

of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For

neither of these an EPAS is likely to exist for the studied parameter :

for Steiner Forest an easy observation shows that the problem is

APX-hard, even if the input graph contains no Steiner vertices. For

Directed Steiner Tree we prove that computing a constant approximation

for this parameter is W[1]-hard. Nevertheless, we

show that an EPAS exists for Unweighted Directed Steiner Tree. Also we

prove that there is an EPAS and a PSAKS for Steiner Forest if in

addition to the number of Steiner vertices, the number of connected

components of an optimal solution is considered to be a parameter.

### lundi 16 octobre 2017, 14h, salle A (2ème étage) : Eunjung Kim (LAMSADE) : Erdos-Posa Property of Chordless Cycles and its Applications

A chordless cycle is a cycle of length at least 4 that has no chord. We prove that the class of all chordless cycles has the Erdos-Posa property, which resolves the major open question concerning the Erdos-Posa property. We complement our main result by showing that the class of all chordless cycles of length at least l for any fixed l ≥ 5 does not have the Erdos-Posa property.

Our proof for chordless cycles is constructive : in polynomial time, one can either find k + 1 vertex-disjoint chordless cycles, or ck2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(opt log opt) for Chordal Vertex Deletion, which improves the best known approximation by Agrawal et. al. The improved approximation algorithm entails improvement over the known kernelization for Chordal Vertex Deletion.

As a corollary, for a non-negative integral function w defined on the vertex set of a graph G, the minimum value \sum_*x\in S* w(x) over all vertex sets S where G − S is forest is at most O(k2 log k) where k is the maximum number of cycles (not necessarily vertex-disjoint) in G such that each vertex v is used at most w(v) times.

### lundi 2 octobre 2017, 14h, P303 : Giuseppe F. Italiano (Universita’ di Roma "Tor Vergata") : 2-Connectivity in Directed Graphs

We survey some recent results on 2-edge and 2-vertex

connectivity in directed graphs. Despite being complete analogs of the

corresponding notions on undirected graphs, in digraphs 2-connectivity

has a much richer and more complicated structure. For undirected

graphs it has been known for over 40 years how to compute all bridges,

articulation points, 2-edge- and 2-vertex-connected components in

linear time, by simply using depth first search. In the case of

digraphs, however, the very same problems have been much more

challenging and have been tackled only very recently.

Exposés de 2016-2017.

Exposés de 2015-2016.

Exposés de 2014-2015.

Pour les exposés antérieurs, voir cette page.