In Directed Geodetic Set, we are given a (directed) graph and seek a small solution set S ⊆ V(G) such that every vertex lies on a shortest directed path between two vertices in S. While most prior work on Directed Geodetic Set has focused on undirected graphs, in this article we study the problem on directed graphs from the perspective of parameterized complexity. It is known that the problem is W[2]-hard when parameterized by the solution size k, even on directed acyclic graphs (DAGs). We investigate structural parameterizations of the problem. Our first result is a kernel of size 2^O(vcn) for Directed Geodetic Set on general digraphs, where vcn denotes the vertex cover number of the underlying (undirected) graph. This implies an algorithm running in time 2^O(vcn²) ⋅ n^O(1). Furthermore, we prove that, assuming the ETH, the problem does not admit an algorithm running in time 2^o(vcn²) ⋅ n^O(1). Such a tight quadratic exponential lower bound in the parameter is relatively uncommon in parameterized complexity. These results generalize earlier work on undirected graphs by Foucaud et al. [STACS 2025], and complements a recent result on directed graph by Foucaud et al. [CALDAM 2026], that showed that the problem is para-NP-hard for the pathwidth and feedback vertex set number of the underlying graph. Next, we show that on general digraphs, Directed Geodetic Set admits a natural kernel of size (kΔ)^O(rdiam), where Δ is the maximum degree and rdiam denotes the reachability diameter of the digraph (a natural analogue of diameter of undirected graphs). This yields an algorithm running in time (kΔ)^O(rdiam⋅k) ⋅ n^O(1). We further prove that, assuming the ETH, the problem does not admit an algorithm running in time (kΔ)^o(rdiam ⋅ k) ⋅ n^O(1). Finally, we justify the necessity of combining parameters by establishing the following hardness results for Directed Geodetic Set: 1) It is W[2]-hard parameterized by k, even on digraphs of maximum degree 3. 2) It is para-NP-hard parameterized by maximum degree and reachability diameter. One can infer that the problem remains W[2]-hard when parameterized by k, even on graphs of reachability diameter 3 from Araújo and Arraes [DAM 2022]. All our conditional lower bounds and hardness results hold even when the input digraph is restricted to be a DAG.

In the Isometric Path Partition problem, the input is a graph G with n vertices and an integer k, and the objective is to determine whether the vertices of G can be partitioned into k vertex-disjoint shortest paths. We investigate the parameterized complexity of the problem when parameterized by the treewidth (tw) of the input graph, arguably one of the most widely studied parameters. Courcelle’s theorem [Information & Computation, 1990] shows that graph problems that are expressible as MSO formulas of constant size admit FPT algorithms parameterized by the treewidth of the input graph. This encompasses many natural graph problems. However, many metric-based graph problems, where the solution is defined using some metric-based property of the graph (often the distance) are not expressible as MSO formulas of constant size. These types of problems, Isometric Path Partition being one of them, require individual attention and often draw the boundary for the success story of parameterization by treewidth. We show that Isometric Path Partition is W[1]-hard when parameterized by treewidth (in fact, even pathwidth (pw)), answering the question by Dumas et al. [SIDMA, 2024], Fernau et al. [TCS, 2025], and confirming the aforementioned tendency. We complement this hardness result by designing a tailored dynamic programming algorithm running in n^{O(tw)} time. This dynamic programming approach also results in an algorithm running in time diam^{O(tw²)} ⋅ n^{O(1)}, where diam is the diameter of the graph. It is known that Isometric Path Partition remains NP-hard on graphs of diameter 2; hence, the combination of both parameters is necessary to obtain a tractable algorithm. Note that the dependency on treewidth is unusually high, as most problems that are FPT for treewidth admit algorithms running in time 2^{O(tw)}⋅ n^{O(1)} or 2^{O(tw log (tw))}⋅ n^{O(1)}. However, we rule out the possibility of a significantly faster algorithm, showing that Isometric Path Partition does not admit an algorithm running in time diam^{o(pw²/(log³(pw)))} ⋅ n^{O(1)}, assuming the Randomized-ETH.

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We study the computational complexity of finding the geodetic number of a graph on chordal graphs and interval graphs. A set $S$ of vertices of a graph $G$ is a \textit{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. We show that \textsc{Minimum Geodetic Set} is fixed parameter tractable for chordal graphs when parameterized by its \emph{tree-width} (which equals its clique number). This implies a polynomial-time algorithm for $k$-trees, for fixed $k$. Then, we show that \textsc{Minimum Geodetic Set} is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012), who showed that \textsc{Minimum Geodetic Set} is polynomial-time solvable on proper interval graphs. As interval graphs are very constrained, to prove the latter result, we design a rather sophisticated reduction technique to work around their inherent linear structure.

We prove that the recognition of exact cubes of trees can be done in polynomial time. More precisely, the exact distance power of a graph is a refinement of the more usual notion of graph power. Given a graph G and a positive integer p , the exact distance p th power of G is the graph G #p on the same vertex set where two vertices are adjacent if their distance is exactly p in G . Recently Bai et al. [Y. Bai, P. P. Cortés, R. Naserasr and D. A. Quiroz. Characterizing and recognizing exact-distance squares of graphs. Discrete Mathematics 347(8). 2024] proved that the recognition of exact squares of trees is polynomially tractable. In order to extend this result to exact cubes of trees, we first test whether there is a caterpillar as an exact cubic root and, if not, proceed with a general tree. Both algorithms rely on the observation that the knowledge of a fixed number of vertices is roughly enough to deduce the whole structure of the tree we aim for.

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\texttt{tw}\cdot \log \texttt{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\texttt{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

For a graph G, a subset S ⊆ V(G) is called a resolving set of G if, for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ 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 ⊆ V(G) of size at most k such that, for any vertex u ∈ V(G), there are two vertices s₁, s₂ ∈ S such that u lies on a shortest path from s₁ to s₂. These two classical problems are known 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. We observe that both problems admit an FPT algorithm running in 2^𝒪(vc²) ⋅ n^𝒪(1) time, and a kernelization algorithm that outputs a kernel with 2^𝒪(vc) vertices, where vc is the vertex cover number. We prove that unless the Exponential Time Hypothesis (ETH) fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, do not admit - an FPT algorithm running in 2^o(vc²) ⋅ n^𝒪(1) time, nor - a kernelization algorithm that does not increase the solution size and outputs a kernel with 2^o(vc) vertices. We only know of one other problem in the literature that admits such a tight algorithmic lower bound with respect to vc. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

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 ⊆ 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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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.

A path cover of a digraph is a collection of paths collectively containing its vertex set. A path cover with minimum cardinality for a directed acyclic graph can be found in polynomial time [Fulkerson, AMS'56; Cáceres et al., SODA'22]. Moreover, Dilworth’s celebrated theorem on chain coverings of partially ordered sets equivalently states that the minimum size of a path cover of a DAG is equal to the maximum size of a set of mutually unreachable vertices. In this paper, we examine how far these classic results can be extended to a dynamic setting. A temporal digraph has an arc set that changes over discrete time-steps; if the underlying digraph is acyclic, then it is 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 path cover is a collection 𝒞 of temporal paths that covers all vertices, and 𝒞 is temporally disjoint if all its temporal paths are pairwise temporally disjoint. We study the computational complexities of the problems of finding a minimum-size temporal (disjoint) path cover (denoted as Temporal Path Cover and Temporally Disjoint Path Cover). On the negative side, we show that both Temporal Path Cover and Temporally Disjoint Path Cover are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, Temporally Disjoint Path Cover remains NP-hard even on temporal oriented trees. We also observe that natural temporal analogues of Dilworth’s theorem on these classes of temporal DAGs do not hold. In contrast, we show that Temporal Path Cover 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, Temporally Disjoint Path Cover becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees. Motivated by the hardness result on trees, we show that, in contrast, Temporal Path Cover admits an XP time algorithm with respect to parameter t_max + tw, where t_max is the maximum time-step and tw is the treewidth of the underlying static undirected graph; moreover, Temporally Disjoint Path Cover admits an FPT algorithm with respect to the same parameterization.

The problems of determining the minimum-sized identifying, locating-dominating and 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) \ 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 graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or 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.

Abstract. We show that if the edges or vertices of an undirected graph [Formula: see text] can be covered by [Formula: see text] shortest paths, then the pathwidth of [Formula: see text] is upper-bounded by a single-exponential function of [Formula: see text]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [Formula: see text] and a set of [Formula: see text] pairs of vertices called terminals, asks whether [Formula: see text] can be covered by [Formula: see text] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [Formula: see text] and a set of [Formula: see text] terminals, asks whether there exist [Formula: see text] shortest paths covering [Formula: see text], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [Formula: see text].

An identifying code of a closed-twin-free graph G is a set S of vertices of G such that any two vertices in G have a distinct intersection between their closed neighborhoods and S. It was conjectured in [F. Foucaud, R. Klasing, A. Kosowski, A. Raspaud. On the size of identifying codes in triangle-free graphs. Discrete Applied Mathematics, 2012] that there exists an absolute constant c such that for every connected graph G of order n and maximum degree ∆, G admits an identifying code of size at most ∆-1/∆n + c. We provide significant support for this conjecture by proving it for the class of all bipartite graphs that do not contain any pairs of open-twins of degree at least 2. In particular, this class of bipartite graphs contains all trees and more generally, all bipartite graphs without 4-cycles. Moreover, our proof allows us to precisely determine the constant c for the considered class, and the list of graphs needing c ≥ 0. For ∆ = 2 (the graph is a path or a cycle), it is long known that c = 3/2 suffices. For connected graphs in the considered graph class, for each ∆ ≥ 3, we show that c = 1/∆ ≤ 1/3 suffices and that c is required to be positive only for a finite number of trees. In particular, for ∆ = 3, there are 12 trees with diameter at most 6 with a positive constant c and, for each ∆ ≥ 4, the only tree with positive constant c is the ∆-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtilä. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022].

A set $S$ of isometric paths of a graph $G$ is ``$v$-rooted'', where $v$ is a vertex of $G$, if $v$ is one of the endpoints of all the isometric paths in $S$. The isometric path complexity of a graph $G$, denoted by $ipco{G}$, is the minimum integer $k$ such that there exists a vertex $v\in V(G)$ satisfying the following property: the vertices of any single isometric path $P$ of $G$ can be covered by $k$ many $v$-rooted isometric paths. First, we provide an $O(n^2 m)$-time algorithm to compute the isometric path complexity of a graph with $n$ vertices and $m$ edges. Then we show that the isometric path complexity remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs. There is a direct algorithmic consequence of having small isometric path complexity. Specifically, we show that if the isometric path complexity of a graph $G$ is bounded by a constant, then there exists a polynomial-time constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose objective is to cover all vertices of a graph with a minimum number of isometric paths. This applies to all the above graph classes.

We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset $S^* \subseteq S$ of minimum cardinality such that for each point $p_i \in P$, the subset $S_i^* \subseteq S^*$ covering $p_i$ satisfies $S_i^*\neq \emptyset$, and each pair $p_i,p_j \in P$, $i \neq j$, we have $S_i^* \neq S_j^*$. In the \emph{continuous version} of the problem, the solution set $S^*$ can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case ($d=1$), the points in $P$ are placed on a horizontal line $L$, and the objects in $S$ are finite-length line segments aligned with $L$ (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. Still, for the 1-dimensional discrete version, we design a polynomial-time $2$-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case ($d=2$) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce $(16\cdot OPT+1)$-approximate and $(64\cdot OPT+1)$-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. We show that the identifying code problem for axis-parallel unit square intersection graphs (in $d=2$) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects.

A path is isometric if it is a shortest path between its endpoints. In this article, we consider the graph covering problem Isometric Path Cover, where we want to cover all the vertices of the graph using a minimum-size set of isometric paths. Although this problem has been considered from a structural point of view (in particular, regarding applications to pursuit-evasion games), it is little studied from the algorithmic perspective. We consider Isometric Path Cover on chordal graphs, and show that the problem is NP-hard for this class. On the positive side, for chordal graphs, we design a 4-approximation algorithm and an FPT algorithm for the parameter solution size. The approximation algorithm is based on a reduction to the classic path covering problem on a suitable directed acyclic graph obtained from a breadth first search traversal of the graph. The approximation ratio of our algorithm is 3 for interval graphs and 2 for proper interval graphs. Moreover, we extend the analysis of our approximation algorithm to k-chordal graphs (graphs whose induced cycles have length at most k) by showing that it has an approximation ratio of k+7 for such graphs, and to graphs of treelength at most 𝓁, where the approximation ratio is at most 6𝓁+2.

We show that if the edges or vertices of an undirected graph G can be covered by k shortest paths, then the pathwidth of G is upper-bounded by a function of k. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph G and a set of k pairs of vertices called terminals, asks whether G can be covered by k shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph G and a set of k terminals, asks whether there exist binom(k,2) shortest paths, each joining a distinct pair of terminals such that these paths cover G). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter k.

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 neighborhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+\ell)/2$, where $n$ is the order and $\ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighborhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$ and characterize the extremal examples as 2-corona graphs of bipartite graphs. This is the best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves to the upper bound $\frac{5n+2\ell}{7}$ when leaves are allowed. This is tight for the 7-cycle $C_7$ and for all stars.

The exact distance p-power of a graph G, denoted G[#p], is a graph on vertex set V(G) in which two vertices are adjacent if they are at distance exactly p in G. Given integers k and p, we define f(k, p) to be the maximum possible order of a clique in the exact distance p-powers of graphs with maximum degree k + 1. It is easily observed that f(k, 2) ≤ k2 + k + 1. We prove that equality may only hold if a connected component of G is isomorphic to a member of the class Pk of incidence graphs of finite projective k-geometries. (These famous combinatorial structures are known to exist when k is a prime power, and are conjectured not to exist for other values of k.) We then study the case of graphs of maximum degree k + 1 with clique number k2 + k. One way to obtain such a graph is to remove a vertex from a graph in P k; we call Pk' the class of all such resulting graphs. We prove that for any graph G of maximum degree k + 1 whose exact square has a (k2 + k)-clique, either G has a subgraph isomorphic to a graph in P’k, or a connected component of G is a (k + 1)-regular bipartite graph of order 2(k2 + k). We call Ok the class of such bipartite graphs, and study their structural properties. These properties imply that (if they exist) the graphs in Ok must be highly symmetric. Using this structural information, we show that O2 contains only one graph, known as the Franklin graph. We then show that O3 also consists of a single graph, which we build. Furthermore, we show that O4 and O5 are empty. For general values of p, we prove that f(k, p) ≤ (k + 1)k[p/2] + 1, and that the bound is tight for every odd integer p ≥ 3. This implies that f(k, 2) = f(k, 3) whenever there exists a finite projective k-geometry, however, in such a case, the bound of f(k, 3) could also be reached by highly symmetric graphs built from a finite k-geometry, which is not the case for other values of k.

A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. A \emph{grid embedding} of a graph is a set of points in two dimensions with integer coordinates such that each point in the set represents a vertex of the graph and, for each edge, the points corresponding to its endpoints are at Euclidean distance~$1$. A graph is a \emph{partial grid} if it has a grid embedding. In this paper, we first prove that \textsc{Minimum Geodetic Set} remains NP-hard even for subcubic partial grids of arbitrary girth. This jointly strengthens three existing hardness results: for bipartite graphs (Dourado et al., Discrete. Math, 2010), subcubic graphs (Bueno et al., Inf. Process. Lett., 2018)~\cite{bueno2018}, and planar graphs (Chakraborty et al., CALDAM, 2020). The \emph{area} of an internal face is the number of integer points lying on the boundary or interior of the face. A graph is a \emph{solid grid} if it has a grid embedding such that all interior faces have area exactly four. To complement the above hardness result, we design a linear-time algorithm for \textsc{Minimum Geodetic Set} on solid grids, improving on a $3$-approximation algorithm by Chakraborty et al. (CALDAM, 2020). Our results hold for \textsc{Edge Geodetic Set} as well. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every edge of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Edge Geodetic Set (MEGS)} problem is to find an edge geodetic set with minimum cardinality of a given graph. As corollaries, we obtain that \textsc{MEGS} remains NP-hard on partial grids and is linear-time solvable on solid grids.

We study two geometric variations of the discriminating code problem. In the discrete version, a finite set of points P and a finite set of objects S are given in ℝ^d. The objective is to choose a subset S^* ⊆ S of minimum cardinality such that the subsets S_i^* ⊆ S^* covering p_i, satisfy S_i^* ≠ ∅ for each i = 1,2,…, n, and S_i^* ≠ S_j^* for each pair (i,j), i ≠ j. In the continuous version, the solution set S^* can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case (d = 1), the points are placed on some fixed-line L, and the objects in S are finite segments of L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in 1D. We then design a polynomial-time 2-approximation algorithm for the 1-dimensional discrete case. We also design a PTAS for both discrete and continuous cases when the intervals are all required to have the same length. We then study the 2-dimensional case (d = 2) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-hard, and design polynomial-time approximation algorithms with factors 4+ε and 32+ε, respectively (for every fixed ε > 0).

A dominating set D in a digraph is a set of vertices such that every vertex is either in D or has an in-neighbour in D. A dominating set D of a digraph is locating-dominating if every vertex not in D has a unique set of in-neighbours within D. The location-domination number γL(G) of a digraph G is the smallest size of a locating-dominating set of G. We investigate upper bounds on γL(G) in terms of the order of G. We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with the same (open or closed) in-neighbourhoods. Thus, we investigate the value of γL(G) in the absence of twins and give a general method for constructing small locating-dominating sets by the means of special dominating sets. In this way, we show that for every twin-free digraph G of order n, γL(G)≤4n5+1 holds, and there exist twin-free digraphs G with γL(G)=2(n−2)3. Improved bounds are proved for certain special cases. In particular, if G is twin-free and a tournament, or twin-free and acyclic, we prove γL(G)≤⌈n2⌉, which is tight in both cases.

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an edge-coloured graph H is a vertex-mapping from G to H that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph H, which generalises the classic vertex-colourability property. The question we are interested in is the following: given an edge-coloured graph G, can we perform k graph operations so that the resulting graph admits a homomorphism to H? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs + and -. We denote the corresponding problems (parameterized by k) by Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring. These problems generalise the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems. Our main focus is on the case where H is an edge-coloured graph of order at most 2, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a PTime/NP-complete complexity dichotomy for all three Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring problems. Then, we address their parameterized complexity. We show that all Vertex Deletion-H-Colouring and Edge Deletion-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless PTime = NP, none of the three considered problems is in XP, since 3-Colouring is NP-complete. We show that the situation is different for Switching-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which Switching-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time f(k)n^{o(k)} for inputs of size n and for any computable function f. For the other cases, Switching-H-Colouring is FPT.

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric dimension $k$. In general, the bound $n\leq d^k+k$ is known to hold. We prove a bound of the form $n=\mathcal{O}(kd^2)$ for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width $w$ and length $\ell$, we obtain a bound of the form $n=\mathcal{O}(kd^2(2\ell+1)^{3w+1})$. This implies in particular that $n=\mathcal{O}(kd^{\mathcal{O}(1)})$ for graphs of constant treewidth and $n=\mathcal{O}(f(k)d^2)$ for chordal graphs, where $f$ is a doubly exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomassé) as a tool, we prove the bounds $n\leq (dk+1)^{t-1}+1$ for $K_t$-minor-free graphs and $n\leq (dk+1)^{d(3\cdot 2^{r}+2)}+1$ for graphs of rankwidth at most $r$.

We study two extremal problems about subgraphs excluding a family $\mathcal{F}$ of graphs: (i) Among all graphs with $m$ edges, what is the smallest size $f(m,\mathcal{F})$ of a largest $\mathcal{F}$-free subgraph? (ii) Among all graphs with minimum degree $\delta$ and maximum degree $\Delta$, what is the smallest minimum degree $h(\delta,\Delta,\mathcal{F})$ of a spanning $\mathcal{F}$-free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where $\mathcal{F}$ is composed of all even cycles of length at most 2r, $r\geq 2$. In this case, we give bounds on $f(m,\mathcal{F})$ and $h(\delta,\Delta,\mathcal{F})$ that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph $G$, we show the existence of subgraphs with arbitrarily high girth and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.

We introduce the notion of a centroidal locating set of a graph G , that is, a set L of vertices such that all vertices in G are uniquely determined by their relative distances to the vertices of L . A centroidal locating set of G of minimum size is called a centroidal basis, and its size is the centroidal dimension . This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph G is lower‐ and upper‐bounded by the metric dimension and twice the location‐domination number of G , respectively. The latter two parameters are standard and well‐studied notions in the field of graph identification. We show that for any graph G with n vertices and maximum degree at least 2, . We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, , the bound being tight for paths and cycles. We finally investigate the computational complexity of determining for an input graph G , showing that the problem is hard and cannot even be approximated efficiently up to a factor of . We also give an ‐approximation algorithm. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(2), 96–108 2014