Pierre Aboulker

• Paris Diderot University

We exhibit a new obstacle to the nascent algorithmic theory for classes excluding an induced minor. We indeed show that on the class of string graphs -- which avoids the 1-subdivision of, say, $K_5$ as an induced minor -- Induced 2-Disjoint Paths is NP-complete. So, while $k$-Disjoint Paths, for a fixed $k$, is polynomial-time solvable in general g...

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree $...

A class of acyclic digraphs $\mathscr{C}$ is linearly unavoidable if there exists a constant $c$ such that every digraph $D\in \mathscr{C}$ is contained in all tournaments of order $c\cdot |V(D)|$. The class of all acyclic digraphs is not linearly avoidable, and Fox, He, and Widgerson recently showed that this is not even the case for acyclic digra...

Given a digraph, an ordering of its vertices defines a \emph{backedge graph}, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The \emph{degreewidth} of a digraph is the minimum over all ordering of the maximum degree of the backedge graph. We answer an open question by Keeney and Lokshtan...

The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies $\vec{\chi}(H) \leq k-1$. We prove various bounds on the minimum number of arcs in a $k$-dicritical digraph, a s...

The dichromatic number of a digraph is the minimum size of a partition of its vertices into acyclic induced subgraphs. Given a class of digraphs , a digraph is a hero in if ‐free digraphs of have bounded dichromatic number. In a seminal paper, Berger et al. give a simple characterisation of all heroes in tournaments. In this paper, we give a simple...

The dichromatic number $\vec\chi(D)$ of a digraph $D$ is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by $\lambda(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara proved that for every digraph $D$, $\vec\chi(D) \leq \lambda(D) + 1$. In this paper, we characterize the digraphs $D$ fo...

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree T...

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obt...

We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every odd integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{3\Delta - 1}{3d - 1} \rce...

We give four new proofs of the directed version of Brook's Theorem and an NP-completeness result.

The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies $\vec{\chi}(H) \leq k-1$. An oriented graph is a digraph with no cycle of length $2$. We prove various bounds on...

The dichromatic number $\dic(D)$ of a digraph $D$ is the least integer $k$ such that $D$ can be partitioned into $k$ directed acyclic digraphs. A digraph is $k$-dicritical if $\dic(D) = k$ and each proper subgraph $D'$ of $D$ satisfies $\dic(D') \leq k-1$. An oriented graph is a digraph with no directed cycle of length $2$. For integers $k$ and $n$...

We show that digraphs with no transitive tournament on $3$ vertices and in which every induced directed cycle has length $3$ can have arbitrarily large dichromatic number. This answers in the negative a question of Carbonero, Hompe, Moore, and Spirkl (and strengthens one of their results).

The dichromatic number of a digraph is the minimum size of a partition of its vertices into acyclic induced subgraphs. Given a class of digraphs $\mathcal C$, a digraph $H$ is a hero in $\mc C$ if $H$-free digraphs of $\mathcal C$ have bounded dichromatic number. In a seminal paper, Berger at al. give a simple characterization of all heroes in tour...

We characterize all digraphs $H$ such that orientations of chordal graphs with no induced copy of $H$ have bounded dichromatic number.

In this paper, we give bounds on the dichromatic number $\vec{\chi}(\Sigma)$ of a surface $\Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $\Sigma$. We determine the asymptotic behaviour of $\vec{\chi}(\Sigma)$ by showing that there exist constants $a_1$ and $a_2$ such that, $ a_1\frac{\sqrt{-c}}{\log(-c)} \leq \...

We show that digraphs with no transitive tournament on $3$ vertices and in which every induced directed cycle has length $3$ can have arbitrarily large dichromatic number. This answers to the negative a question of Carbonero, Hompe, Moore, and Spirkl (and extends some of their results).

We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{\Delta}{d} \rceil, d)$-edge c...

A set of n $n$ points in the plane which are not all collinear defines at least n $n$ distinct lines. Chen and Chvátal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced c...

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K4)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary. We p...

We give four new proofs of the directed version of Brook's Theorem and an NP-completeness result.

We study the dichromatic number of a digraph, defined as the minimum number of parts in a partition of its vertex set into acyclic induced subdigraphs. We consider the class of oriented graphs such that the out-neighbourhood of any vertex induces a transitive tournament and prove for it a decomposition theorem. As a consequence, we obtain that orie...

In this paper, we give bounds on the dichromatic number χ→(Σ) of a surface Σ, which is the maximum dichromatic number of an oriented graph embeddable on Σ. We determine the asymptotic behaviour of χ→(Σ) by showing that there exist constants a1 and a2 such that, a1-clog(-c)≤χ→(Σ)≤a2-clog(-c) for every surface Σ with Euler characteristic c≤-2. We the...

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture....

In this paper, we give bounds on the dichromatic number $\vec{\chi}(\Sigma)$ of a surface $\Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $\Sigma$. We show that there exist two constants $a_1$ and $a_2$ such that, $ a_1\frac{\sqrt{-c}}{\log(-c)} \leq \vec{\chi}(\Sigma) \leq a_2 \sqrt{\frac{-c}{\log(-c)}} $ for e...

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gy\'arf\'as-Sumner conjecture. More...

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019]. The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary. W...

A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chv\'atal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle o...

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity...

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity...

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity...

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescen...

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescen...

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space. We prove that in any metric space with $n$ points, either ther...

A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chv\'atal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work we prove a slightly stronger version of Chen and Chv\'atal conjecture for a family of graphs...

A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chv\'atal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work we prove a slightly stronger version of Chen and Chv\'atal conjecture for a family of graphs...

A famous conjecture of Gy\'arf\'as and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for...

A famous conjecture of Gy\'arf\'as and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for...

A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at most $k$. We prove Brooks-type theorems for $k$-connected graphs with maximal local edge-connectivity $k$, and for any graph with maximal local edge-connectivity 3. We also consider several relat...

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of...

A 4‐wheel is a graph formed by a cycle C and a vertex not in C that has at least four neighbors in C. We prove that a graph G that does not contain a 4‐wheel as a subgraph is 4‐colorable and we describe some structural properties of such a graph.

In [{Structural properties and decomposition of linear balanced matrices}, {\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also...

A special case of a theorem of De Bruijn and Erd\H{o}s asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced b...

A \emph{wheel} is a graph formed by a chordless cycle $C$ and a vertex $u$ not in $C$ that has at least three neighbors in $C$. We prove that every 3-connected planar graph that does not contain a wheel as an induced subgraph is either a line graph or has a clique cutset. We prove that every planar graph that does not contain a wheel as an induced...

A wheel is a graph formed by a chordless cycle and a vertex that has at least three neighbors in the cycle. We prove that every 3-connected graph that does not contain a wheel as a subgraph is in fact minimally 3-connected. We give a new proof of a theorem of Thomassen and Toft: every graph that does not contain a wheel as a subgraph is 3-colorable...

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least...

Chen and Chv\'atal introduced the notion of lines in hypergraphs; they proved that every 3-uniform hypergraph with $n$ vertices either has a line that consists of all $n$ vertices or else has at least $\log_2 n$ distinct lines. We improve this lower bound by a factor of $2-o(1)$.

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of...

We show how the notion of a moplex, related to LexBFS and first defined by Berry and Bordat, can be used to design fast algorithms for solving problems in several classes of graph, namely even-hole-free graphs, wheel-free graphs and universally signable graphs.

We recall several known results about minimally 2-connected graphs and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least...

We show how the notion of a moplex, related to LexBFS and first defined by Berry and Bordat, can be used to design fast algorithms for solving problems in several classes of graph, such as even-hole-free graphs and universally signable graphs.