# Florent FoucaudUniversité Clermont Auvergne | Univ BPC · LIMOS Laboratoire d’Informatique, de Modélisation et d’Optimisation des Systèmes

PhD

99
Publications
4,984
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657
Citations
Citations since 2017
60 Research Items
501 Citations
January 2021 - present
Position
• Maître de conférences
September 2019 - December 2020
Position
• PostDoc Position
Description
• Homomorphisms of signed graphs
May 2019 - July 2019
Position
• Teaching staff

## Publications

Publications (99)
Preprint
Full-text available
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...
Preprint
Full-text available
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 sets of colors found in their neighborhoods are different. The neighbor-locating chromatic number $\chi_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-coloring of...
Chapter
For a graph G=(V,E) with a vertex set V and an edge set E, a function f:V→{0,1,2,...,diam(G)} is called a broadcast on G. For each vertex u∈V, if there exists a vertex v in G (possibly, u=v) such that f(v)>0 and d(u,v)≤f(v), then f is called a dominating broadcast on G. The cost of the dominating broadcast f is the quantity ∑v∈Vf(v). The minimum co...
Chapter
A proper k-vertex-coloring of a graph G is a neighbor-locatingk-coloring if for each pair of vertices in the same color class, the sets of colors found in their neighborhoods are different. The neighbor-locating chromatic number χNL(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 l...
Chapter
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...
Chapter
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...
Preprint
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We give essentially tight bounds for, $\nu(d,k)$, 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 $\nu(d,k) \leqslant \exp(\exp(O(d)))k$,...
Preprint
Full-text available
The \emph{isometric path antichain cover number} of a graph $G$, denoted by $ipacc(G)$, is a graph parameter that was recently introduced to provide a constant factor approximation algorithm for \textsc{Isometric Path Cover}, whose objective is to cover all vertices of a graph with a minimum number of isometric paths (i.e. shortest paths between th...
Preprint
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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 for short) has been studied extensively, as it includes Hamiltonian Path as a special case. However, the natural variants where the paths are required to be either induced, called Induced Path Partitio...
Article
Full-text available
We study geometric variations of the discriminating code problem. In the 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 $$S_i^* \subseteq S^*$$ cov...
Preprint
Full-text available
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 $\gamma^L_t(G)$. It has...
Preprint
Full-text available
We introduce the Red-Blue Separation problem on graphs, where we are given a graph $G=(V,E)$ whose vertices are colored either red or blue, and we want to select a (small) subset $S \subseteq V$, called red-blue separating set, such that for every red-blue pair of vertices, there is a vertex $s \in S$ whose closed neighborhood contains exactly one...
Preprint
Full-text available
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...
Article
We introduce a new graph-theoretic concept in the area of network monitoring. A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge e of G, there are a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. We denote by dem(G) the smallest size of such a set in G. The vertices of...
Preprint
Full-text available
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 \emph{terminals}, asks whether $G$ c...
Preprint
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 $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of...
Article
Full-text available
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 edg...
Preprint
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 $C$. 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 tree...
Chapter
We introduce the Red-Blue Separation problem on graphs, where we are given a graph $$G = (V, E)$$ whose vertices are colored either red or blue, and we want to select a (small) subset $$\mathcal{S} \subseteq V$$, called red-blue separating set, such that for every red-blue pair of vertices, there is a vertex $$s \in \mathcal S$$ whose closed neighb...
Preprint
Full-text available
A \emph{signed graph} $(G, \sigma)$ is a graph $G$ together with an assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between the theories of minors and colorings of graphs. Following this thread of thoughts, we investigate this connec...
Preprint
Full-text available
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 $\sigma= v_1,\dots,v_n$, the First-Fit'' greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighb...
Article
An open neighbourhood locating-dominating set is a set S of vertices of a graph G such that each vertex of G has a neighbour in S, and for any two vertices u,v of G, there is at least one vertex in S that is a neighbour of exactly one of u and v. We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of...
Article
An injective k-edge-coloring of a graph G is an assignment of colors, i.e. integers in {1,…,k}, to the edges of G such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a k-coloring exists is called Injective k-Edge-Coloring. We show that Injective 3-Edge-Co...
Article
Full-text available
We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph D is a function f:V(D)→N such that for each vertex v of D, there exists a vertex t with f(t)>0 having a directed path to v of length at most f(t). The cost of f is the sum of f(v...
Article
Given a graph H, a graph G is called H-critical if G does not admit a homomorphism to H, but any proper subgraph of G does. Observe that Kk−1-critical graphs are the standard k-(colour)-critical graphs. We consider questions of extremal nature previously studied for k-critical graphs and generalize them to H-critical graphs. After complete graphs,...
Preprint
An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a $k$-coloring exists is called k-INJECTIVE EDGE-COLORING. We show that...
Article
Full-text available
We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted χ[♯2](G). This notion is related to other t...
Preprint
An open neighbourhood locating-dominating set is a set $S$ of vertices of a graph $G$ such that each vertex of $G$ has a neighbour in $S$, and for any two vertices $u,v$ of $G$, there is at least one vertex in $S$ that is a neighbour of exactly one of $u$ and $v$. We characterize those graphs whose only open neighbourhood locating-dominating set is...
Article
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)...
Preprint
We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set} if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of $G$ such that $e$ belongs to all shortest paths between $x$ and $y$. We denote by $dem(G)$ the smallest size of suc...
Preprint
We study two geometric variations of the discriminating code problem. In the \emph{discrete version}, 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 the subsets $S_i^* \subseteq S^*$ covering $p_i$, satisfy $S_i^*\neq \emp... Preprint Full-text available We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted$\chi^{\sharp 2}(G)$. This notion is relate... Preprint Full-text available We study the complexity of finding the \emph{geodetic number} on subclasses of planar graphs and chordal graphs. A set$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 car... Conference Paper We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph D is a function $$f:V(D)\rightarrow \mathbb {N}$$ such that for each vertex v of D, there exists a vertex t with $$f(t)>0$$ having a directed path to v of length at most f(t). Th... Article We study homomorphism problems of signed graphs from a computational point of view. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept when studying signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a... Preprint We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph$D$is a function$f:V(D)\to\mathbb{N}$such that for each vertex$v$of$D$, there exists a vertex$t$with$f(t)>0$having a directed path to$v$of length at most$f(t)$. The... Article 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... Chapter In this paper, we study the computational complexity of finding the geodetic number of graphs. A set of vertices S of a graph G is a geodetic set if any vertex of G lies in some shortest path between some pair of vertices from S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality. In this paper, we prove that... Chapter We introduce a new graph-theoretic concept in the area of network monitoring. A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge e of G, there is a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. We denote by the smallest size of such a set in G. The vertices of M repres... Chapter Given a graph H, a graph G is called H-critical if G does not admit a homomorphism to H, but any proper subgraph of G does. Observe that -critical graphs are the classic k-(colour)-critical graphs. This work is a first step towards extending questions of extremal nature from k-critical graphs to H-critical graphs. Besides complete graphs, the next... Preprint 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$\gamma_L(G)$of a digraph$G$is the smallest size of a locati... Preprint We study the complexity of graph modification problems for homomorphism-based 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 gra... Preprint Full-text available In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices$S$of a graph$G$is a \emph{geodetic set} if any vertex of$G$lies in some shortest path between some pair of vertices from$S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality... Preprint Full-text available We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencie... Article A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty–Simon conjecture, states that any diameter-2-critical graph of order n has at most ⌊n2∕4⌋ edges, with equality if and only if G is a balanced complete bipartite graph. Many partial results about this... Chapter A graph database is a digraph whose arcs are labelled with symbols from a fixed alphabet. A regular graph pattern (RGP) is a digraph whose edges are labelled with regular expressions over the alphabet. RGPs model navigational queries for graph databases, more precisely, conjunctive regular path queries. A match of a navigational RGP query in the da... Preprint A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty-Simon conjecture, states that any diameter-2-critical graph of order n has at most n${}^2$/4 edges, with equality if and only if G is a balanced complete bipartite graph. Many partial results about th... Article A signed graph$(G, \Sigma)$is a graph$G$and a subset$\Sigma$of its edges which corresponds to an assignment of signs to the edges: edges in$\Sigma$are negative while edges not in$\Sigma$are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced othe... Article We introduce the problem PARTIAL VC DIMENSION that asks, given a hypergraph H=(X,E) and integers k and ℓ whether one can select a set C⊆X of k vertices of H such that the set {e∩C,e∈E} of distinct hyperedge-intersections with C has size at least ℓ. The sets e∩C define equivalence classes over E. PARTIAL VC DIMENSION is a generalization of VC DIMENS... Article In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical graph domination where selected vertices may have different domination ranges. The minimum cost of a dominating broadcast in a graph$G$is denoted by$\gamma_b(G)$. When formulated as an integer linear program, the dual of broadcast domination is called multipac... Article The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is n... Article The goal of this work is to study homomorphism problems (from a computational point of view) on two superclasses of graphs:$2$-edge-coloured graphs and signed graphs. On the one hand, we consider the$H$-COLOURING problem when$H$is a$2$-edge-coloured graph, and we show that a dichotomy theorem would imply the dichotomy conjecture of Feder and V... Article An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with... Article We present a necessary and sufficient condition for a graph of odd-girth$2k+1$to bound the class of$K_4$-minor-free graphs of odd-girth (at least)$2k+1$, that is, to admit a homomorphism from any such$K_4$-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every$K_4$-m... Article Full-text available Gr\"otzsch introduced his famous graph as a triangle-free graph which is not 3-colourable. Harary showed that any such graph must be of order at least 11 and Chv\'atal proved that the Gr\"otzsch graph is the only such graph on 11 vertices. As a generalization, Ngoc and Tuza asked about the order of a smallest graph of odd-girth$2k+1$which is not... Article Full-text available 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 fo... Conference Paper We study the problems Locating-Dominating Set and Metric Dimension, which consist of determining a minimum-size set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. We prove that both decision problems are NP-complet... Conference Paper We introduce the problem Partial VC Dimension that asks, given a hypergraph$H=(X,E)$and integers$k$and$\ell$, whether one can select a set$C\subseteq X$of$k$vertices of$H$such that the set$\{e\cap C, e\in E\}$of distinct hyperedge-intersections with$C$has size at least$\ell$. The sets$e\cap C$define equivalence classes over$E$. P... Article Full-text available A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. A locating-total dominating set of G is a total dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u) ∩ D ≠ N(v)... Article A tropical graph$(H,c)$consists of a graph$H$and a (not necessarily proper) vertex-colouring$c$of$H$. Given two tropical graphs$(G,c_1)$and$(H,c)$, a homomorphism of$(G,c_1)$to$(H,c)$is a standard graph homomorphism of$G$to$H$that also preserves the vertex-colours. We initiate the study of the computational complexity of tropical... Article Full-text available We study the complexity of the problem DETECTION PAIR. A detection pair of a graph$G$is a pair$(W,L)$of sets of detectors with$W\subseteq V(G)$, the watchers, and$L\subseteq V(G)$, the listeners, such that for every pair$u,v$of vertices that are not dominated by a watcher of$W$, there is a listener of$L$whose distances to$u$and to$v$... Article A set$D$of vertices of a graph$G$is locating every two distinct vertices outside$D$have distinct neighbors in$D$, that is, for distinct vertices$u$and$v$outside$D$,$N(u) \cap D \ne N(v) \cap D$where$N(u)$denotes the open neighborhood of$u$. If$D$is also a dominating set (total dominating set), it is called a locating-dominating s... Article Full-text available It was proved by Raychaudhuri in 1987 that if a graph power$G^{k-1}$is an interval graph, then so is the next power$G^k$. This result was extended to$m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of$G^{k-1}$can be extended to an interval representation of$G^k$t... Article A dominating set of a graph$G$is a set$D$of vertices of$G$such that every vertex outside$D$is adjacent to a vertex in$D$. A locating-dominating set of$G$is a dominating set$D$of$G$with the additional property that every two distinct vertices outside$D$have distinct neighbors in$D$; that is, for distinct vertices$u$and$v$outsid... Article A locating-dominating set a of graph$G$is a dominating set$D$of$G$with the additional property that every two distinct vertices outside$D$have distinct neighbors in$D$; that is, for distinct vertices$u$and$v$outside$D$,$N(u) \cap D \ne N(v) \cap D$where$N(u)$denotes the open neighborhood of$u$. A graph is twin-free if every two d... Article 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$CD(G)$. This notion, wh... Article An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number... Article Full-text available Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identi... Conference Paper Full-text available The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We show that Grundy Coloring can be solved in time $$O^*(2.443^n)$$ on g... Article We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph using a subset of the vertices and either the neighborhood within the solution set or the distances to the solution vertices. Usin... Article We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph using a subset of the vertices and either the neighborhood within the solution set or the distances to the solution vertices. Usin... Conference Paper A signed graph (G, Σ) is an undirected graph G together with an assignment of signs (positive or negative) to all its edges, where Σ denotes the set of negative edges. Two signatures are said to be equivalent if one can be obtained from the other by a sequence of resignings (i.e. switching the sign of all edges incident to a given vertex). Extendin... Article We study two extremal problems about subgraphs excluding a family$\mathcal{F}$of fixed 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,\Delt...
Article
A graph G=(V,E) is arbitrarily partitionable if for any sequence @t of positive integers adding up to |V|, there is a sequence of vertex-disjoint subsets of V whose orders are given by @t, and which induce connected subgraphs. Such a graph models, e.g., a computer network which may be arbitrarily partitioned into connected subnetworks. In this pape...
Article
This paper introduces the problem of identifying vertices of a graph using paths. An identifying path cover of a graph G is a set PP of paths such that each vertex belongs to a path of PP, and for each pair u, v of vertices, there is a path of PP which includes exactly one of u, v. This new notion is related to a large number of other identificatio...
Article
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code problem in line graphs. If $\ID(G)$ denotes the size of a minimum identifying code of an identifiable graph $G$, we...