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## Multi angle  Tree decompositions and graph algorithms Lokshtanov, Daniel (Auteur de la Conférence) | CIRM (Editeur )

A central concept in graph theory is the notion of tree decompositions - these are decompositions that allow us to split a graph up into "nice" pieces by "small" cuts. It is possible to solve many algorithmic problems on graphs by decomposing the graph into "nice" pieces, finding a solution in each of the pieces, and then gluing these solutions together to form a solution to the entire graph. Examples of this approach include algorithms for deciding whether a given input graph is planar, the $k$-Disjoint paths algorithm of Robertson and Seymour, as well as many algorithms on graphs of bounded tree-width. In this talk we will look at a way to compare two tree decompositions of the same graph and decide which of the two is "better". It turns out that for every cut size $k$, every graph $G$ has a tree decomposition with (approximately) this cut size, such that this tree-decomposition is "better than" every other tree-decomposition of the same graph with cut size at most $k$. We will discuss some consequences of this result, as well as possible improvements and research directions. A central concept in graph theory is the notion of tree decompositions - these are decompositions that allow us to split a graph up into "nice" pieces by "small" cuts. It is possible to solve many algorithmic problems on graphs by decomposing the graph into "nice" pieces, finding a solution in each of the pieces, and then gluing these solutions together to form a solution to the entire graph. Examples of this approach include algorithms for ...

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## Multi angle  Coloring graphs with forbidden induced paths Chudnovsky, Maria (Auteur de la Conférence) | CIRM (Editeur )

The problem of testing if a graph can be colored with a given number $k$ of colors is $NP$-complete for every $k>2$. But what if we have more information about the input graph, namely that some fixed graph $H$ is not present in it as an induced subgraph? It is known that the problem remains $NP$-complete even for $k=3$, unless $H$ is the disjoint union of paths. We consider the following two questions: 1) For which graphs $H$ is there a polynomial time algorithm to 3-color (or in general $k$-color) an $H$-free graph? 2) For which graphs $H$ are there finitely many 4-critical $H$-free graphs? This talk will survey recent progress on these questions, and in particular give a complete answer to the second one. The problem of testing if a graph can be colored with a given number $k$ of colors is $NP$-complete for every $k>2$. But what if we have more information about the input graph, namely that some fixed graph $H$ is not present in it as an induced subgraph? It is known that the problem remains $NP$-complete even for $k=3$, unless $H$ is the disjoint union of paths. We consider the following two questions: 1) For which graphs $H$ is there a ...

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## Multi angle  Restricted types of matchings Rautenbach, Dieter (Auteur de la Conférence) | CIRM (Editeur )

We present new results concerning restricted types of matchings such as uniquely restricted matchings and acyclic matchings, and we also consider the corresponding edge coloring notions. Our focus lies on bounds, exact and approximative algorithms. Furthermore, we discuss some matching removal problems. The talk is based on joined work with J. Baste, C. Lima, L. Penso, I. Sau, U. Souza, and J. Szwarcfiter.

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## Multi angle  Le problème Graph Motif - Partie 2 Fertin, Guillaume (Auteur de la Conférence) | CIRM (Editeur )

Le problème Graph Motif est défini comme suit : étant donné un graphe sommet colorié G=(V,E) et un multi-ensemble M de couleurs, déterminer s'il existe une occurrence de M dans G, c'est-à-dire un sous ensemble V' de V tel que
(1) le multi-ensemble des couleurs de V' correspond à M,
(2) le sous-graphe G' induit par V' est connexe.
Ce problème a été introduit, il y a un peu plus de 10 ans, dans le but de rechercher des motifs fonctionnels dans des réseaux biologiques, comme par exemple des réseaux d'interaction de protéines ou des réseaux métaboliques. Graph Motif a fait depuis l'objet d'une attention particulière qui se traduit par un nombre relativement élevé de publications, essentiellement orientées autour de sa complexité algorithmique.
Je présenterai un certain nombre de résultats algorithmiques concernant le problème Graph Motif, en particulier des résultats de FPT (Fixed-Parameter Tractability), ainsi que des bornes inférieures de complexité algorithmique.
Ceci m'amènera à détailler diverses techniques de preuves dont certaines sont plutôt originales, et qui seront je l'espère d'intérêt pour le public.
Le problème Graph Motif est défini comme suit : étant donné un graphe sommet colorié G=(V,E) et un multi-ensemble M de couleurs, déterminer s'il existe une occurrence de M dans G, c'est-à-dire un sous ensemble V' de V tel que
(1) le multi-ensemble des couleurs de V' correspond à M,
(2) le sous-graphe G' induit par V' est connexe.
Ce problème a été introduit, il y a un peu plus de 10 ans, dans le but de rechercher des motifs fonctionnels dans des ...

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## Multi angle  Embedding extension problems Mohar, Bojan (Auteur de la Conférence) | CIRM (Editeur )

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## Multi angle  Bidimensionality and subexponential parameterized algorithms Thilikos, Dimitrios (Auteur de la Conférence) | CIRM (Editeur )

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## Multi angle  Induced cycles and coloring Chudnovsky, Maria (Auteur de la Conférence) | CIRM (Editeur )

A hole in a graph is an induced cycle of length at least four, and an odd hole is a hole of odd length. A famous conjecture of A. Gyárfás [1] from 1985 asserts:
Conjecture 1: For all integers $k,l$ there exists $n(k,l)$ such that every graph $G$ with no clique of carnality more than $k$ and no odd hole of length more than $l$ has chromatic number at most $n(k,l)$.

In other words, the conjecture states that the family of graphs with no long odd holes is $\chi$-bounded. Little progress was made on this problem until recently Scott and Seymour proved that Conjecture 1 is true for all pairs $(k,l)$ when $l=3$ (thus excluding all odd holes guarantees $\chi$-boundedness) [3].
No other cases have been settled, and here we focus on the case $k=2$. We resolve the first open case, when $k=2$ and $l=5$, proving that
Theorem 1. Every graph with no triangle and no odd hole of length $>5$ is $82200$-colorable.

Conjecture 1 has a number of other interesting special cases that still remain open; for instance
* Conjecture: For all $l$ every triangle-free graph $G$ with sufficiently large chromatic number has an odd hole of length more than $l$;
* Conjecture: For all $k,l$ every graph with no clique of size more than $k$ and sufficiently large chromatic number has a hole of length more than $l$.

We prove both these statements with the additional assumption that $G$ contains no $5$-hole. (The latter one was proved, but not published, by Scott earlier, improving on [2]). All the proofs follows a similar outline. We start with a leveling of a graph with high chromatic number, that is a classification of the vertices by their distance from a fixed root. Then the graph undergoes several rounds of "trimming" that allows us to focus on a subgraph $M$ with high chromatic number that is, in some sense, minimal. We also ensure that certain pairs of vertices with a neighbor in $M$ can be joined by a path whose interior is anticomplete to $M$. It is now enough to find two long paths between some such pair of vertices, both with interior in $M$ and of lengths of different parity, to obtain a long odd hole.
A hole in a graph is an induced cycle of length at least four, and an odd hole is a hole of odd length. A famous conjecture of A. Gyárfás [1] from 1985 asserts:
Conjecture 1: For all integers $k,l$ there exists $n(k,l)$ such that every graph $G$ with no clique of carnality more than $k$ and no odd hole of length more than $l$ has chromatic number at most $n(k,l)$.

In other words, the conjecture states that the family of graphs with no long odd ...