Lucas Pastor
Laboratoire G-SCOP, bureau B521
46 avenue Félix Viallet
38031 Grenoble Cedex 1
France

Telephone: +33 (0) 4 56 52 98 22
Mail: lucas.pastor@grenoble-inp.fr

I'm a third year PhD student in the Combinatorial Optimization team of G-SCOP. My advisors are Frédéric Maffray (G-SCOP) and Sylvain Gravier (Institut Fourier).
I'm working on graph theory. I am interested in about everything in this field even though my main works are about graph coloring and structural graph theory.

# Decomposition techniques applied to the Clique-Stable set Separation problem

with Nicolas Bousquet, Aurélie Lagoutte and Frédéric Maffray
arXiv:1703.07106

In a graph, a Clique-Stable Set separator (CS-separator) is a family $\mathcal{C}$ of cuts (bipartitions of the vertex set) such that for every clique $K$ and every stable set $S$ with $K \cap S = \emptyset$, there exists a cut $( W,W')$ in $\mathcal{C}$ such that $K \subseteq W$ and $S \subseteq W'$. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [17] asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive and negative results related to this question were given recently. Here we show how graph decomposition can be used to prove that a class of graphs admits a polynomial CS-separator. We apply this method to apple-free graphs and cap-free graphs.

# Maximum Weight Stable Set in ($P_7$, bull)-free graphs and ($S_{1,2,3}$, bull)-free graphs

with Frédéric Maffray
arXiv:1611.09663

We give a polynomial time algorithm that finds the maximum weight stable set in a graph that does not contain an induced path on seven vertices or a bull (the graph with vertices $a$, $b$, $c$, $d$, $e$ and edges $ab$, $bc$, $cd$, $be$, $ce$). With the same arguments with also give a polynomial algorithm for any graph that does not contain $S_{1,2,3}$ or a bull.

# Disproving the normal graph conjecture

with Ararat Harutyunyan and Stéphan Thomassé
arXiv:1508.05487

A graph $G$ is called normal if there exist two coverings, $\mathbb{C}$ and $\mathbb{S}$ of its vertex set such that every member of $\mathbb{C}$ induces a clique in $G$, every member of $\mathbb{S}$ induces an independent set in $G$ and $C \cap S \neq \emptyset$ for every $C \in \mathbb{C}$ and $S \in \mathbb{S}$. It has been conjectured by De Simone and Körner in 1999 that a graph $G$ is normal if $G$ does not contain $C_5$, $C_7$ and $\overline{C_7}$ as an induced subgraph. We disprove this conjecture.

# Colouring squares of claw-free graphs

with Rémi de Joannis de Verclos and Ross J. Kang
arXiv:1609.08645
To appear in Canadian Journal of Mathematics

Is there some absolute $\varepsilon > 0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\chi(G^2) \le (2-\varepsilon) \omega(G)^2$, where $\omega(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case of $G$ the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erdős and Nešetřil.

# The maximum weight stable set problem in ($P_6$, bull)-free graphs

with Frédéric Maffray
arXiv:1602.06817 $-$ doi:10.1007/978-3-662-53536-3_8 $-$ WG 2016
Lecture Notes in Computer Science 9941 (2016), 85-96

We present a polynomial-time algorithm that finds a maximum weight stable set in a graph that does not contain as an induced subgraph an induced path on six vertices or a bull (the graph with vertices $a, b, c, d, e$ and edges $ab, bc, cd, be, ce$).

# On the choosability of claw-free perfect graphs

with Sylvain Gravier and Frédéric Maffray
arXiv:1506.02576 $-$ doi:10.1007/s00373-016-1732-9
Graphs and Combinatorics 32:6 (2016), 2393-2413

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most $4$.

# $4$-coloring ($P_6$, bull)-free graphs

with Frédéric Maffray
arXiv:1511.08911 $-$ doi:10.1016/j.dam.2016.03.011
Discrete Applied Mathematics (2016) $-$ To appear

We present a polynomial-time algorithm that determines whether a graph that contains no induced path on six vertices and no bull (the graph with vertices $a,b,c,d,e$ and edges $ab,bc,cd,be,ce$) is $4$-colorable. We also show that for any fixed $k$ the $k$-coloring problem can be solved in polynomial time in the class of $(P_6,\mbox{bull},\mbox{gem})$-free graphs.

## Talks

• Présentation AGIR, UFR IM2AG (Grenoble, France) November 24 2016. Coloration dans les graphes structurés. Slides.

• Présentation JGA, Université Paris-Dauphine (Paris, France) November 16 2016. Indépendant de poids maximum dans les ($P_6$, bull)-free. Slides.

• Séminaire LIMOS, (Clermont-Ferrand, France) October 12 2016. Disproving the normal graph conjecture. Slides.

• Séminaire LaBRI, équipe Graphes et Optimisation (Bordeaux, France) March 4 2016. The coloring problem in graphs with structural restrictions. Slides.

• Séminaire G-SCOP (Grenoble, France) October 1st 2015. Coloration par liste dans les graphes parfaits sans griffe. Slides.

• Présentation JGA, (Orléans, France) November 4 2015. Coloration par liste dans les graphes parfaits sans griffe. Slides.

• Third STINT meeting (Autrans, France) June 30 2015. List-coloring in claw-free perfect graphs. Slides.

• Présentation Journées G-SCOP, (Loriol, France) June 4 2015. Coloration par liste. Slides.

## Teaching

• 2016-2017 Modélisation des structures informatiques (Modelisation of computer science structures), L2, TD/TP, Université Joseph Fourier, Grenoble.

• 2016-2017 Base de données et Bases de Connaissances, L3, TD/TP, UFR Informatique, Mathématiques, Mathématiques Appliquées, Grenoble.

• 2015-2016 Introduction à UNIX et à la programmation en langage C (introduction to UNIX and C programming), L1, TD/TP, Université Joseph Fourier, Grenoble.

• 2014-2015 Informatique instrumentale et multimédia (introduction to algorithm, programming and HTML/CSS), L1, TD/TP, Université Joseph Fourier, Grenoble.

## Other

##### Research projects:
• I'm a member of the ANR project STINT.