# Dimitrios M. Thilikos's research while affiliated with French National Centre for Scientific Research and other places

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## Publications (343)

We introduce an annotated extension of treewidth that measures the contribution of a vertex set $X$ to the treewidth of a graph $G.$ This notion provides a graph distance measure to some graph property $\mathcal{P}$: A vertex set $X$ is a $k$-treewidth modulator of $G$ to $\mathcal{P}$ if the treewidth of $X$ in $G$ is at most $k$ and its removal g...

We introduce a graph-parametric framework for obtaining obstruction characterizations of graph parameters with respect to partial ordering relations. For this, we define the notions of class obstruction, parametric obstruction, and universal obstruction as combinatorial objects that determine the asymptotic behavior of graph parameters. Our framewo...

The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph c...

We introduce the notion of universal obstruction of a graph parameter, with respect to some quasi-ordering relation. Universal obstructions may serve as compact characterizations of the asymptotic behavior of graph parameters. We provide order-theoretic conditions which imply that such a characterization is finite and, when this is the case, we pre...

A strict bramble of a graph G is a collection of pairwise-intersecting connected subgraphs of G. The order of a strict bramble \({{\mathcal {B}}}\) is the minimum size of a set of vertices intersecting all sets of \({{\mathcal {B}}}.\) The strict bramble number of G, denoted by \(\textsf{sbn}(G),\) is the maximum order of a strict bramble in G. The...

Disjoint-paths logic, denoted FO+DP, extends first-order logic (FO) with atomic predicates dp_k[(x_1,y_1),...,(x_k,y_k)], expressing the existence of internally vertex-disjoint paths between x_i and y_i, for 1<=i<=k. We prove that for every graph class excluding some fixed graph as a topological minor, the model checking problem for FO+DP is fixed-...

For a finite collection of graphs \(\mathcal {F} \) , the \(\mathcal {F} \) -TM-Deletion problem has as input an n -vertex graph G and an integer k and asks whether there exists a set S ⊆ V ( G ) with | S | ≤ k such that G ∖ S does not contain any of the graphs in \(\mathcal {F} \) as a topological minor. We prove that for every such \(\mathcal {F}...

\noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $\#\mathsf{P}$-hard. In 1913 P\'olya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the bia...

In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property \(\mathcal {P} \) . Typically, the modification operation ⊠ may be vertex deletion , edge deletion , edge contraction , or edge addition and the question is, given a graph G and an integer k , whether it is possible to transform G to...

The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate ${\sf dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in\{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential...

Let ${\cal G}$ be a minor-closed graph class and let $G$ be an $n$-vertex graph. We say that $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. Our first result is an algorithm that decides whether $G$ is a $k$-apex of ${\cal G}$ in time $2^{{\sf poly}(k)}\cdot n^2$, whe...

We introduce the parameter of block elimination distance as a measure of how close a graph is to some particular graph class. Formally, given a graph class \({\mathcal {G}}\), the class \({\mathcal {B}}({\mathcal {G}})\) contains all graphs whose blocks belong to \({\mathcal {G}}\) and the class \({\mathcal {A}}({\mathcal {G}})\) contains all graph...

Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $\Gamma_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has treewidth $\mathcal{O}({\bf bcg}(G)^{c})$ for some $1\leq c<2$. The SQG${\bf C}$ property is important for algorithm...

We introduce a new kernelization tool, called rainbow matching technique, that is appropriate for the design of polynomial kernels for packing problems. Our technique capitalizes on the powerful combinatorial results of [Graf, Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on two (di)graph packing problems, namely the Triangle-...

A linkage in a graph $G$ of size $k$ is a subgraph $L$ of $G$ whose connected components are $k$ paths. The pattern of a linkage of size $k$ is the set of $k$ pairs formed by the endpoints of these paths. A consequence of the Unique Linkage Theorem is the following: there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that if a plane graph $G$...

The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every $t\in\mathbb{N},$ there exists some constant $c_{t}$ such that every $K_{t}$-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most $c_{t}$ vertices, to graphs that can be seen as the union of some grap...

We consider the mixed search game against an agile and visible fugitive. This is the variant of the classic fugitive search game on graphs where searchers may be placed to (or removed from) the vertices or slide along edges. Moreover, the fugitive resides on the edges of the graph and can move at any time along unguarded paths. The mixed search num...

The elimination distance to some target graph property \(\color {MidnightBlack}\mathcal {P} \) is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem’s fixed-parameter tractability by identifying sufficient...

Let \(\mathcal {G} \) be a minor-closed graph class. We say that a graph \(G \) is a \(k \) -apex of \(\mathcal {G} \) if \(G \) contains a set \(S \) of at most \(k \) vertices such that \(G\setminus S \) belongs to \(\mathcal {G} \) . We denote by \(\mathcal {A}_k (\mathcal {G}) \) the set of all graphs that are \(k \) -apices of \(\mathcal {G}....

Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γk. A graph class G has the SQGC property if every graph G∈G has treewidth O(bcg(G)c) for some 1≤c<2. The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results,...

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ is the maximum order of a strict bramble in $G.$ The strict bramble n...

We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of edge-treewidth. We first observe that edge-treewidth does not enjoy any closeness properties under the known partia...

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. We propose a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the...

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-...

We introduce the parameter of block elimination distance as a measure of how close a graph is to some particular graph class. Formally, given a graph class \(\mathcal{G}\), the class \(\mathcal{B}(\mathcal{G})\) contains all graphs whose blocks belong to \(\mathcal{G}\) and the class \(\mathcal{A}(\mathcal{G})\) contains all graphs where the remova...

We introduce the rendezvous game with adversaries. In this game, two players, Facilitator and Divider, play against each other on a graph. Facilitator has two agents, and Divider has a team of k agents located in some vertices of the graph. They take turns in moving their agents to adjacent vertices (or staying put). Facilitator wins if his agents...

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model a...

In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to tr...

The {\sc Weighted} $\mathcal{F}$-\textsc{Vertex Deletion} for a class ${\cal F}$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$ such that $G-S\in{\cal F}.$ The case when ${\cal F}$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for \t...

In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness. 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 (2015) as a...

The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the struct...

It is well known that the treewidth of a graph G corresponds to the node search number where a team of searchers is pursuing a fugitive that is lazy and invisible (or alternatively is agile and visible) and has the ability to move with infinite speed via unguarded paths. Recently, monotone and connected node search strategies have been considered....

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION (resp. ${\cal F}$-TM-DELETION) problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor (resp. topological minor). We are int...

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor. We are interested in the parameterized complexity of ${\cal F}$-M-D...

We introduce the block elimination distance as a measure of how close a graph is to some particular graph class. Formally, given a graph class ${\cal G}$, the class ${\cal B}({\cal G})$ contains all graphs whose blocks belong to ${\cal G}$ and the class ${\cal A}({\cal G})$ contains all graphs where the removal of a vertex creates a graph in ${\cal...

Let ${\cal G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}.$ We denote by ${\cal A}_k ({\cal G})$ the set of all graphs that are $k$-apices of ${\cal G}.$ We prove that every graph in the obstruction set of ${\cal A...

We introduce the rendezvous game with adversaries. In this game, two players, {\sl Facilitator} and {\sl Disruptor}, play against each other on a graph. Facilitator has two agents, and Disruptor has a team of $k$ agents located in some vertices of the graph. They take turns in moving their agents to adjacent vertices (or staying). Facilitator wins...

The node‐search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search‐game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search gam...

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in...

The concept of compactor has been introduced in Kim et al. (2018) as a general data-reduction concept for parametrized counting problems. For a function F:Σ∗→N and a parameterization κ:Σ∗→N, a compactor (C,E) consists of a polynomial-time computable function P, called condenser, and a computable function M, called extractor, such that F=M∘P. If the...

Neural networks are the pinnacle of artificial intelligence , as in recent years we witnessed many novel architectures, learning and optimization techniques for deep learning. As neural networks inherently constitute multipartite graphs among neuron layers, we aim to analyze directly their structure to extract meaningful information from the learni...

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L (and subclasses of it), wit...

We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most $s$ unblocked edges ($s$ can be seen as the speed of the robber). Both parts...

A subgraph complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class \({\mathscr {G}}\), is there a subgraph complement of G which is in \({\mathscr {G}}\)? We show that this problem can be solved in polynomial...

We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges (s can be seen as the speed of the robber). Both parts hav...

For a finite fixed collection of graphs F, the F-M-Deletion problem consists in, given a graph G and an integer k, decide whether there exists S⊆V(G) with |S|≤k such that G∖S does not contain any of the graphs in F as a minor. We provide lower bounds under the ETH on the smallest function fF such that F-M-Deletion can be solved in time fF(tw)⋅nO(1)...

Let ${\cal G}$ be a graph class. We say that a graph $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. We prove that if ${\cal G}$ is minor-closed, then there is an algorithm that either returns a set $S$ certifying that $G$ is a $k$-apex of ${\cal G}$ or reports that s...

The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search where we are given a system of tunnels that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a search...

In parameterized complexity, a kernelization algorithm can be seen as a reduction of a parameterized problem to itself, so that the produced equivalent instance has size depending exclusively on the parameter. If this size is polynomial, then we say that the parameterized problem in question admits a polynomial kernelization algorithm. Kernelizatio...

Neural networks are the pinnacle of Artificial Intelligence, as in recent years we witnessed many novel architectures, learning and optimization techniques for deep learning. Capitalizing on the fact that neural networks inherently constitute multipartite graphs among neuron layers, we aim to analyze directly their structure to extract meaningful i...

A graph is sub-unicyclic if it contains at most one cycle. A graph G is k-apex sub-unicyclic if it can become sub-unicyclic by removing k of its vertices. We identify 29 graphs that are the minor-obstructions of the class of 1-apex sub-unicyclic graphs. For bigger values of k, we give an exact structural characterization of all the cactus graphs th...

It is well known that the treewidth of a graph $G$ corresponds to the node search number where a team of cops is pursuing a robber that is lazy, visible and has the ability to move at infinite speed via unguarded path. In recent papers, connected node search strategies have been considered. A search stratregy is connected if at each step the set of...

For a finite collection of graphs F, the F-M-Deletion (resp. F-TM-Deletion) problem consists in, given a graph G and an integer k, decide whether there exists S⊆V(G) with |S|≤k such that G∖S does not contain any of the graphs in F as a minor (resp. topological minor). We are interested in the parameterized complexity of both problems when the param...

We establish connections between parameterized/kernelization complexity of graph modification problems and expressibility in logic. For a first-order logic formula φ, we consider the problem of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the resulting modification has the property expressibl...

Given a finite set of graphs H and a non-negative integer k, we define Ak(H) as the set containing every graph G that has k vertices whose removal provides a graph without any of the graphs in H as a minor. It is known that if H contains at least one planar graph then each obstruction in Ak(H) has at most kcH vertices, for some cH depending only on...

This book constitutes the revised papers of the 45th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2019, held in Vall de Núria, Spain, in June 2019.
The 29 full papers presented in this volume were carefully reviewed and selected from 87 submissions. They cover a wide range of areas, aiming at connecting theory and appl...

The Disjoint Paths problem asks whether a fixed number of pairs of terminals in a graph $G$ can be linked by pairwise disjoint paths. In the context of this problem, Robertson and Seymour introduced the celebrated irrelevant vertex technique that has since become standard in graph algorithms. The technique consists of detecting a vertex that is irr...

For a fixed connected graph $H$, the $\{H\}$-M-DELETION problem asks, given a graph $G$, for the minimum number of vertices that intersect all minor models of $H$ in $G.$ It is known that this problem can be solved in time $f(tw)\cdot n^{O(1)}$, where $tw$ is the treewidth of $G.$ We determine the asymptotically optimal function $f(tw),$ for each p...

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-TM-Deletion problem has as input an $n$-vertex graph $G$ and an integer $k$ and asks whether there exists a set $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a topological minor. We prove that for every such ${\cal F}$,...

During the last years, several algorithmic meta-theorems have appeared (Bodlaender et al. [FOCS 2009], Fomin et al. [SODA 2010], Kim et al. [ICALP 2013]) guaranteeing the existence of linear kernels on sparse graphs for problems satisfying some generic conditions. The drawback of such general results is that it is usually not clear how to derive fr...

A graph is sub-unicyclic if it contains at most one cycle. We also say that a graph $G$ is $k$-apex sub-unicyclic if it can become sub-unicyclic by removing $k$ of its vertices. We identify 29 graphs that are the minor-obstructions of the class of $1$-apex sub-unicyclic graphs, i.e., the set of all minor minimal graphs that do not belong in this cl...

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 th...

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model a...

A pseudoforest is a graph whose number of edges is at most its number of vertices. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests.

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model a...

We study the concept of \emph{compactor}, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function $F:\Sigma^*\to \Bbb{N}$ and a parameterization $\kappa: \Sigma^*\to \Bbb{N}$, a compactor $({\sf P},{\sf M})$ consists of a polynomial-time computable function ${\sf P}$, called \emph{condenser}, a...

The notion of tree-cut width has been introduced by Wollan [The structure of graphs not admitting a fixed immersion. {\it Journal of Combinatorial Theory, Series B}, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions ca...

We study the class L of link types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with...

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with...

We consider the problems of deciding whether an input graph can be modified by removing/adding at most $k$ vertices/edges such that the result of the modification satisfies some property definable in first-order logic. We establish a number of sufficient and necessary conditions on the quantification pattern of the first-order formula $\phi$ for th...

A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a partial complement of $G$ which is in $\mathcal{G}$? We show that this problem can be solved in polynomial tim...

We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In {\sc Clustering to Given Connectivities}, we are given an $n$-vertex graph $G$, an integer $k$, and a sequence $\Lambda=\langle \lambda_{1},\ldots,\lambda_{t}\rangle$ of positive integers and we ask whether it is po...

We give the first linear kernels for the Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor. In other words, we prove the existence of polynomial time algorithms that, for a given H-topological-minor-free graph G and a positive integer k, output an H-topological-minor-free graph G′ on O(k...

We prove that the branchwidth of a 2-edge-connected hypergraph embedded on an oriented 2-manifold of genus g is an (1, 2g)-self-dual parameter, partially resolving a conjecture by Sau and Thilikos.