arXiv subjects

Jean-Florent Raymond

Publications and source records attributed to Jean-Florent Raymond.

At least 37 records · Page 2Linked to original sources

arXiv2020

A Menger-like property of tree-cut width

In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness [A Menger-like property of tree-width: The finite case. Journal of Combinatorial Theory, Series B, 48(1):67-76, 1990]. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan as a possible edge-version of tree decompositions [The structure of graphs not admitting a fixed immersion. Journal of Combinatorial Theory, Series B, 110:47-66, 2015]. We show that every graph admits a tree-cut decomposition of minimum width that additionally satisfies an edge-connectivity condition analogous to Thomas' leanness.

math.CO↗

arXiv2020

Enumerating minimal dominating sets in $K_t$-free graphs and variants

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for $K_t$-free graphs and variants. This answers a question of Kanté et al. about enumeration in bipartite graphs.

cs.DM↗

arXiv2020

On the tractability of optimization problems on H-graphs

For a graph $H$, a graph $G$ is an $H$-graph if it is an intersection graph of connected subgraphs of some subdivision of $H$. $H$-graphs naturally generalize several important graph classes like interval or circular-arc graph. This class was introduced in the early 1990s by Bíró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of $H$-graphs by showing that a number of fundamental optimization problems are solvable in polynomial time on $H$-graphs. We extend and complement these algorithmic findings in several directions. First we show that for every fixed $H$, the class of $H$-graphs is of logarithmically-bounded boolean-width (via mim-width). Pipelined with the plethora of known algorithms on graphs of bounded boolean-width, this describes a large class of problems solvable in polynomial time on $H$-graphs. We also observe that $H$-graphs are graphs with polynomially many minimal separators. Combined with the work of Fomin, Todinca and Villanger on algorithmic properties of such classes of graphs, this identify another wide class of problems solvable in polynomial time on $H$-graphs. The most fundamental optimization problems among the problems solvable in polynomial time on $H$-graphs are Maximum Clique, Maximum Independent Set, and Minimum Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of Parameterized Complexity. We show that Maximum Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by the size of $H$ plus the size of the solution. On the other hand, we prove that when $H$ is a tree, then Minimum Dominating Set is fixed-parameter tractable parameterized (FPT) by the size of $H$. For Maximum Clique we show that it admits a polynomial kernel parameterized by $H$ and the solution size.

cs.DS↗

arXiv2019

A tight Erdős-Pósa function for planar minors

Let $H$ be a planar graph. By a classical result of Robertson and Seymour, there is a function $f:\mathbb{N} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$ and all graphs $G$, either $G$ contains $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or there is a subset $X$ of at most $f(k)$ vertices such that $G-X$ has no $H$-minor. We prove that this remains true with $f(k) = c k \log k$ for some constant $c=c(H)$. This bound is best possible, up to the value of $c$, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with $f(k) = c k \log^d k$ for some universal constant $d$. The proof is constructive and yields a polynomial-time $O(\log \mathsf{OPT})$-approximation algorithm for packing subgraphs containing an $H$-minor.

math.CO↗

arXiv2018

Packing and covering induced subdivisions

A class $\mathcal{F}$ of graphs has the induced Erdős-Pósa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cycle $C_{\ell}$ of length $\ell$, the class of $C_{\ell}$-subdivisions has the induced Erdős-Pósa property if and only if $\ell\le 4$. In this paper, we investigate whether or not the class of $H$-subdivisions has the induced Erdős-Pósa property for other graphs $H$. We completely settle the case when $H$ is a forest or a complete bipartite graph. Regarding the general case, we identify necessary conditions on $H$ for the class of $H$-subdivisions to have the induced Erdős-Pósa property. For this, we provide three basic constructions that are useful to prove that the class of the subdivisions of a graph does not have the induced Erdős-Pósa property. Among remaining graphs, we prove that if $H$ is either the diamond, the $1$-pan, or the $2$-pan, then the class of $H$-subdivisions has the induced Erdős-Pósa property.

cs.DM↗

arXiv2018

A tight Erdős-Pósa function for wheel minors

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.

cs.DM↗

arXiv2018

Induced minors and well-quasi-ordering

A graph $H$ is an induced minor of a graph $G$ if it can be obtained from an induced subgraph of $G$ by contracting edges. Otherwise, $G$ is said to be $H$-induced minor-free. Robin Thomas showed that $K_4$-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without $K_4$ and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 -- 247, 1985]. We provide a dichotomy theorem for $H$-induced minor-free graphs and show that the class of $H$-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if $H$ is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990].

math.CO↗

arXiv2018

Hitting minors, subdivisions, and immersions in tournaments

The Erdős-Pósa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph $H$ (resp. strongly-connected directed graph $H$), the class of directed graphs that contain $H$ as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erdős-Pósa property in the class of tournaments. We also prove that if $H$ is a strongly-connected directed graph, the class of directed graphs containing $H$ as an immersion has the edge-Erdős-Pósa property in the class of tournaments.

cs.DM↗

arXiv2017

Multigraphs without large bonds are wqo by contraction

We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental.

math.CO↗

arXiv2017

Polynomial expansion and sublinear separators

Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed $0<δ\le 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $O(n^{1-δ})$, then any depth-$k$ minor (i.e. minor obtained by contracting disjoint subgraphs of radius at most $k$) of a graph in $\mathcal{C}$ has average degree $O\big((k \text{ polylog }k)^{1/δ}\big)$. This confirms a conjecture of Dvořák and Norin.

math.CO↗

arXiv2017

Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.

cs.DS↗

arXiv2017

An $O(\log OPT)$-approximation for covering and packing minor models of $θ_r$

Given two graphs $G$ and $H$, we define $\textsf{v-cover}_{H}(G)$ (resp. $\textsf{e-cover}_{H}(G)$) as the minimum number of vertices (resp. edges) whose removal from $G$ produces a graph without any minor isomorphic to ${H}$. Also $\textsf{v-pack}_{H}(G)$ (resp. $\textsf{v-pack}_{H}(G)$) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of $G$ that contain a minor isomaorphic to $H$. We denote by $θ_r$ the graph with two vertices and $r$ parallel edges between them. When $H=θ_r$, the parameters $\textsf{v-cover}_{H}$, $\textsf{e-cover}_{H}$, $\textsf{v-pack}_{H}$, and $\textsf{v-pack}_{H}$ are NP-hard to compute (for sufficiently big values of $r$). Drawing upon combinatorial results in [Minors in graphs of large $θ_r$-girth, Chatzidimitriou et al., arXiv:1510.03041], we give an algorithmic proof that if $\textsf{v-pack}_{θ_r}(G)\leq k$, then $\textsf{v-cover}_{θ_r}(G) = O(k\log k)$, and similarly for $\textsf{v-pack}_{θ_r}$ and $\textsf{e-cover}_{θ_r}$. In other words, the class of graphs containing ${θ_r}$ as a minor has the vertex/edge Erdős-Pósa property, for every positive integer $r$. Using the algorithmic machinery of our proofs, we introduce a unified approach for the design of an $O(\log {\rm OPT})$-approximation algorithm for $\textsf{v-pack}_{θ_r}$, $\textsf{v-cover}_{θ_r}$, $\textsf{v-pack}_{θ_r}$, and $\textsf{e-cover}_{θ_r}$ that runs in $O(n\cdot \log(n)\cdot m)$ steps. Also, we derive several new Erdős-Pósa-type results from the techniques that we introduce.

cs.DS↗

arXiv2017

Minors in graphs of large $θ_r$-girth

For every $r \in \mathbb{N}$, let $θ_r$ denote the graph with two vertices and $r$ parallel edges. The $θ_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $θ_r$. This notion generalizes the usual concept of girth which corresponds to the case $r=2$. In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213--225, 2003], Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of $θ_{r}$-girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed $r$, graphs excluding as a minor the disjoint union of $k$ $θ_{r}$'s have treewidth $O(k\cdot \log k)$.

math.CO↗

arXiv2016

Well-quasi-ordering H-contraction-free graphs

A well-quasi-order is an order which contains no infinite decreasing sequence and no infinite collection of incomparable elements. In this paper, we consider graph classes defined by excluding one graph as contraction. More precisely, we give a complete characterization of graphs H such that the class of H-contraction-free graphs is well-quasi-ordered by the contraction relation. This result is the contraction analogue on the previous dichotomy theorems of Damsaschke [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427-435, 1990] on the induced subgraph relation, Ding [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489-502, 1992] on the subgraph relation, and Błasiok et al. [Induced minors and well-quasi-ordering, ArXiv e-prints, 1510.07135, 2015] on the induced minor relation.

math.CO↗

arXiv2016

Recent techniques and results on the Erdős-Pósa property

Several min-max relations in graph theory can be expressed in the framework of the Erdős-Pósa property. Typically, this property reveals a connection between packing and covering problems on graphs. We describe some recent techniques for proving this property that are related to tree-like decompositions. We also provide an unified presentation of the current state of the art on this topic.

cs.DM↗

arXiv2016

Linear kernels for edge deletion problems to immersion-closed graph classes

Suppose $\mathcal{F}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion of at most $k$ edges of $G$ can result in a graph that does not contain any graph from $\mathcal{F}$ as an immersion. This problem is a close relative of the $\mathcal{F}$-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from $\mathcal{F}$. We prove that whenever all graphs from $\mathcal{F}$ are connected and at least one graph of $\mathcal{F}$ is planar and subcubic, then the $\mathcal{F}$-Immersion Deletion problem admits: a constant-factor approximation algorithm running in time $O(m^3 \cdot n^3 \cdot \log m)$; a linear kernel that can be computed in time $O(m^4 \cdot n^3 \cdot \log m)$; and a $O(2^{O(k)} + m^4 \cdot n^3 \cdot \log m)$-time fixed-parameter algorithm, where $n,m$ count the vertices and edges of the input graph. These results mirror the findings of Fomin et al. [FOCS 2012], who obtained a similar set of algorithmic results for $\mathcal{F}$-Minor Deletion, under the assumption that at least one graph from $\mathcal{F}$ is planar. An important difference is that we are able to obtain a linear kernel for $\mathcal{F}$-Immersion Deletion, while the exponent of the kernel of Fomin et al. for $\mathcal{F}$-Minor Deletion depends heavily on the family $\mathcal{F}$. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of $\mathcal{F}$-Immersion Deletion is quite different than that of $\mathcal{F}$-Minor Deletion.

cs.DS↗

arXiv2016

Scattered packings of cycles

We consider the problem Scattered Cycles which, given a graph $G$ and two positive integers $r$ and $\ell$, asks whether $G$ contains a collection of $r$ cycles that are pairwise at distance at least $\ell$. This problem generalizes the problem Disjoint Cycles which corresponds to the case $\ell = 1$. We prove that when parameterized by $r$, $\ell$, and the maximum degree $Δ$, the problem Scattered Cycles admits a kernel on $24 \ell^2 Δ^\ell r \log(8 \ell^2 Δ^\ell r)$ vertices. We also provide a $(16 \ell^2 Δ^\ell)$-kernel for the case $r=2$ and a $(148 Δr \log r)$-kernel for the case $\ell = 1$. Our proofs rely on two simple reduction rules and a careful analysis.

cs.DM↗

arXiv2016

Packing and Covering Immersion Models of Planar subcubic Graphs

A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected planar subcubic graph on $h>0$ edges, $G$ is a graph, and $k$ is a non-negative integer, then either $G$ contains $k$ vertex/edge-disjoint subgraphs, each containing $H$ as an immersion, or $G$ contains a set $F$ of $f(k,h)$ vertices/edges such that $G\setminus F$ does not contain $H$ as an immersion.

math.CO↗