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Florent Foucaud

Publications and source records attributed to Florent Foucaud.

At least 37 records · Page 2Linked to original sources

arXiv2025

Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove the (tight) conditional lower bounds for these problems when parameterized by treewidth and solution as. Formally, \textsc{Locating-Dominating Set} (respectively, \textsc{Test Cover}) parameterized by the treewidth of the input graph (respectively, of the natural auxiliary graph) does not admit an algorithm running in time $2^{2^{o(tw)}} \cdot poly(n)$ (respectively, $2^{2^{o(tw)}} \cdot poly(|U| + |\mathcal{F}|))$. This result augments the small list of NP-Complete problems that admit double exponential lower bounds when parameterized by treewidth. Then, we first prove that \textsc{Locating-Dominating Set} does not admit an algorithm running in time $2^{o(k^2)} \cdot poly(n)$, nor a polynomial time kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(k)}$ vertices, unless the Ð fails. To the best of our knowledge, \textsc{Locating-Dominating Set} is the first problem that admits such an algorithmic lower-bound (with a quadratic function in the exponent) when parameterized by the solution size. Finally, we prove that \textsc{Test Cover} does not admit an algorithm running in time $2^{2^{o(k)}} \cdot poly(|U| + |\mathcal{F}|)$. This is also a rare example of the problem that admits a double exponential lower bound when parameterized by the solution size. We also present algorithms whose running times match the above lower bounds.

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arXiv2025

Monitoring edge-geodetic sets in graphs

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor the vertices and/or the edges of a network (viewed as a graph) in order to detect and prevent failures. Inspired by two notions studied in the literature (edge-geodetic sets and distance-edge-monitoring sets), we define the notion of a monitoring edge-geodetic set (MEG-set for short) of a graph $G$ as an edge-geodetic set $S\subseteq V(G)$ of $G$ (that is, every edge of $G$ lies on some shortest path between two vertices of $S$) with the additional property that for every edge $e$ of $G$, there is a vertex pair $x, y$ of $S$ such that $e$ lies on all shortest paths between $x$ and $y$. The motivation is that, if some edge $e$ is removed from the network (for example if it ceases to function), the monitoring probes $x$ and $y$ will detect the failure since the distance between them will increase. We explore the notion of MEG-sets by deriving the minimum size of a MEG-set for some basic graph classes (trees, cycles, unicyclic graphs, complete graphs, grids, hypercubes, corona products...) and we prove an upper bound using the feedback edge set of the graph. We also show that determining the smallest size of an MEG-set of a graph is NP-hard, even for graphs of maximum degree at most~9.

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arXiv2025

Algorithms and complexity for monitoring edge-geodetic sets in graphs

A monitoring edge-geodetic set of a graph is a subset $M$ of its vertices such that for every edge $e$ in the graph, deleting $e$ increases the distance between at least one pair of vertices in $M$. We study the following computational problem \textsc{MEG-set}: given a graph $G$ and an integer $k$, decide whether $G$ has a monitoring edge geodetic set of size at most $k$. We prove that the problem is NP-hard even for 2-apex 3-degenerate graphs, improving a result by Haslegrave (Discrete Applied Mathematics 2023). Additionally, we prove that the problem cannot be solved in subexponential-time, assuming the Exponential-Time Hypothesis, even for 3-degenerate graphs. Further, we prove that the optimization version of the problem is APX-hard, even for 4-degenerate graphs. Complementing these hardness results, we prove that the problem admits a polynomial-time algorithm for interval graphs, a fixed-parameter tractable algorithm for general graphs with clique-width plus diameter as the parameter, and a fixed-parameter tractable algorithm for chordal graphs with treewidth as the parameter. We also provide an approximation algorithm with factor $\ln m\cdot OPT$ and $\sqrt{n\ln m}$ for the optimization version of the problem, where $m$ is the number of edges, $n$ the number of vertices, and $OPT$ is the size of a minimum monitoring edge-geodetic set of the input graph.

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arXiv2025

Problems in NP can Admit Double-Exponential Lower Bounds when Parameterized by Treewidth or Vertex Cover

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $Σ_2^p$, $Π_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

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arXiv2025

Metric Dimension and Geodetic Set Parameterized by Vertex Cover

For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

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arXiv2025

Characterizing optimal monitoring edge-geodetic sets for some structured graph classes

Given a graph $G=(V,E)$, a set $S\subseteq V$ is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in $S$. The minimum size of such a set in $G$ is called the monitoring edge-geodetic number of $G$ and is denoted by $meg(G)$. In this work, we compute the monitoring edge-geodetic number efficiently for the following graph classes: distance-hereditary graphs, $P_4$-sparse graphs, bipartite permutation graphs, and strongly chordal graphs. The algorithms follow from structural characterizations of the optimal monitoring edge-geodetic sets for these graph classes in terms of \emph{mandatory vertices} (those that need to be in every solution). This extends previous results from the literature for cographs, interval graphs and block graphs.

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arXiv2025

Monitoring arc-geodetic sets of oriented graphs

Monitoring edge-geodetic sets in a graph are subsets of vertices such that every edge of the graph must lie on all the shortest paths between two vertices of the monitoring set. These objects were introduced in a work by Foucaud, Krishna and Ramasubramony Sulochana with relation to several prior notions in the area of network monitoring like distance edge-monitoring. In this work, we explore the extension of those notions unto oriented graphs, modelling oriented networks, and call these objects monitoring arc-geodetic sets. We also define the lower and upper monitoring arc-geodetic number of an undirected graph as the minimum and maximum of the monitoring arc-geodetic number of all orientations of the graph. We determine the monitoring arc-geodetic number of fundamental graph classes such as bipartite graphs, trees, cycles, etc. Then, we characterize the graphs for which every monitoring arc-geodetic set is the entire set of vertices, and also characterize the solutions for tournaments. We also cover some complexity aspects by studying two algorithmic problems. We show that the problem of determining if an undirected graph has an orientation with the minimal monitoring arc-geodetic set being the entire set of vertices, is NP-hard. We also show that the problem of finding a monitoring arc-geodetic set of size at most $k$ is $NP$-complete when restricted to oriented graphs with maximum degree $4$.

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arXiv2025

Bounds and extremal graphs for monitoring edge-geodetic sets in graphs

A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring edge-geodetic number of $G$, denoted by $meg(G)$, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem. In this article, we compare $meg(G)$ with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs $G$ that have $V(G)$ as their minimum MEG-set, which settles an open problem due to Foucaud \textit{et al.} (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for $meg(G)$ for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of $G$. We examine the change in $meg(G)$ with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.

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arXiv2024

Parameterizing Path Partitions

We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W[1]-hard on DAGs. We also show that SPP is in XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results transfer to the case of covering by edge-disjoint paths, and purely covering.

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arXiv2024

Structural Parameterization of Locating-Dominating Set and Test Cover

We investigate structural parameterizations of two identification problems: LOCATING-DOMINATING SET and TEST COVER. In the first problem, an input is a graph $G$ on $n$ vertices and an integer $k$, and one asks if there is a subset $S$ of $k$ vertices such that any two distinct vertices not in $S$ are dominated by distinct subsets of $S$. In the second problem, an input is a set of items $U$, a set of subsets $\mathcal{F}$ of $U$ called $tests$ and an integer $k$, and one asks if there is a set $S$ of at most $k$ tests such that any two items belong to distinct subsets of tests of $S$. These two problems are "identification" analogues of DOMINATING SET and SET COVER, respectively. Chakraborty et al. [ISAAC 2024] proved that both the problems admit conditional double-exponential lower bounds and matching algorithms when parameterized by treewidth of the input graph. We continue this line of investigation and consider parameters larger than treewidth, like vertex cover number and feedback edge set number. We design a nontrivial dynamic programming scheme to solve TEST COVER in "slightly super-exponential" time $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ in the number $|U|$ of items and LOCATING-DOMINATING SET in time $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$, where $\textsf{vc}$ is the vertex cover number and $n$ is the order of the graph. This shows that the lower bound results with respect to treewidth from Chakraborty et al. [ISAAC 2024] cannot be extended to vertex cover number. We also show that, parameterized by feedback edge set number, LOCATING-DOMINATING SET admits a linear kernel thereby answering an open question in [Cappelle et al., LAGOS 2021]. Finally, we show that neither LOCATING-DOMINATING SET nor TEST COVER is likely to admit a compression algorithm returning an input with a subquadratic number of bits, unless $\textsf{NP} \subseteq \textsf{coNP}/poly$.

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arXiv2024

Progress towards the two-thirds conjecture on locating-total dominating sets

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $γ^L_t(G)$. It has been conjectured that $γ^L_t(G)\leq\frac{2n}{3}$ holds for every twin-free graph $G$ of order $n$ without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.

math.CO↗

arXiv2024

On locating and neighbor-locating colorings of sparse graphs

A proper $k$-coloring of a graph $G$ is a \emph{neighbor-locating $k$-coloring} if for each pair of vertices in the same color class, the two sets of colors found in their respective neighborhoods are different. The \textit{neighbor-locating chromatic number} $χ_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-vertex-coloring of a graph $G$ is a \emph{locating $k$-coloring} if for each pair of vertices $x$ and $y$ in the same color-class, there exists a color class $S_i$ such that $d(x,S_i)\neq d(y,S_i)$. The locating chromatic number $χ_{L}(G)$ is the minimum $k$ for which $G$ admits a locating $k$-coloring. Our main results concern the largest possible order of a sparse graph of given neighbor-locating chromatic number. More precisely, we prove that if $G$ has order $n$, neighbor-locating chromatic number $k$ and average degree at most $2a$, where $2a\le k-1$ is a positive integer, then $n$ is upper-bounded by $\mathcal{O}(a^2(k^{2a+1}))$. We also design a family of graphs of bounded maximum degree whose order is close to reaching this upper bound. Our upper bound generalizes two previous bounds from the literature, which were obtained for graphs of bounded maximum degree and graphs of bounded cycle rank, respectively. Also, we prove that determining whether $χ_L(G)\le k$ and $χ_{NL}(G)\le k$ are NP-complete for sparse graphs: more precisely, for graphs with average degree at most 7, maximum average degree at most 20 and that are $4$-partite. We also study the possible relation between the ordinary chromatic number, the locating chromatic number and the neighbor-locating chromatic number of a graph.

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arXiv2024

On three domination-based identification problems in block graphs

The problems of determining the minimum-sized \emph{identifying}, \emph{locating-dominating} and \emph{open locating-dominating codes} of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set $C$ of a graph $G$ such that the vertices of a chosen subset of $V(G)$ (i.e. either $V(G)\setminus C$ or $V(G)$ itself) are uniquely determined by their neighborhoods in $C$. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graphs classes. In this work, we present tight lower and upper bounds for all three types of codes for \emph{block graphs} (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or \emph{blocks}) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature - with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

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arXiv2024

Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $σ= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $σ$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $χ(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $χ(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.

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arXiv2024

Algorithms and complexity for path covers of temporal DAGs: when is Dilworth dynamic?

In this paper, we study a dynamic analogue of the Path Cover problem, which can be solved in polynomial-time in directed acyclic graphs. A temporal digraph has an arc set that changes over discrete time-steps, if the underlying digraph (the union of all the arc sets) is acyclic, then we have a temporal DAG. A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal (resp. temporally disjoint) path cover is a collection of (resp. temporally disjoint) temporal paths that covers all vertices. In this paper, we study the computational complexities of the problems of finding a temporal (disjoint) path cover with minimum cardinality, denoted as Temporal Path Cover (TPC) and Temporally Disjoint Path Cover (TD-PC). We show that both problems are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, TD-PC remains NP-hard even on temporal oriented trees. In contrast, we show that TPC is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, TD-PC becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees. We also show that TPC (resp. TD-PC) admits an XP (resp. FPT) time algorithm with respect to parameter tmax + tw, where tmax is the maximum time-step, and tw is the treewidth of the underlying static undirected graph.

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arXiv2024

Extremal digraphs for open neighbourhood location-domination and identifying codes

A set $S$ of vertices of a digraph $D$ is called an open neighbourhood locating-dominating set if every vertex in $D$ has an in-neighbour in $S$, and for every pair $u,v$ of vertices of $D$, there is a vertex in $S$ that is an in-neighbour of exactly one of $u$ and $v$. The smallest size of an open neighbourhood locating-dominating set of a digraph $D$ is denoted by $γ_{OL}(D)$. We study the class of digraphs $D$ whose only open neighbourhood locating-dominating set consists of the whole set of vertices, in other words, $γ_{OL}(D)$ is equal to the order of $D$. We call those digraphs extremal. By considering digraphs with loops allowed, our definition also applies to the related (and more widely studied) concept of identifying codes. We extend previous studies from the literature for both open neighbourhood locating-dominating sets and identifying codes of both undirected and directed graphs. These results all correspond to studying open neighbourhood locating-dominating sets on special classes of digraphs. To do so, we prove general structural properties of extremal digraphs, and we describe how they can all be constructed. We then use these properties to give new proofs of several known results from the literature. We also give a recursive and constructive characterization of the extremal di-trees (digraphs whose underlying undirected graph is a tree).

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arXiv2023

Neighbourhood complexity of graphs of bounded twin-width

We give essentially tight bounds for, $ν(d,k)$, the maximum number of distinct neighbourhoods on a set $X$ of $k$ vertices in a graph with twin-width at most~$d$. Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet et al., Algorithmica '22; Przybyszewski '22] have shown the upper bound $ν(d,k) \leqslant \exp(\exp(O(d)))k$, with a double-exponential dependence in the twin-width. The work of [Gajarsky et al., ICALP '22], using the framework of local types, implies the existence of a single-exponential bound (without explicitly stating such a bound). We give such an explicit bound, and prove that it is essentially tight. Indeed, we give a short self-contained proof that for every $d$ and $k$ $$ν(d,k) \leqslant (d+2)2^{d+1}k = 2^{d+\log d+Θ(1)}k,$$ and build a bipartite graph implying $ν(d,k) \geqslant 2^{d+\log d+Θ(1)}k$, in the regime when $k$ is large enough compared to~$d$.

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arXiv2023

Bounds and extremal graphs for total dominating identifying codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $γ^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $γ_t^{\text{ID}}(G)$. Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $γ_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $γ_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $γ^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached). Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains this bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins). Finally, we relate $γ_t^{\text{ID}}(G)$ to the related parameter $γ^{\text{ID}}(G)$ as well as the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.

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