A locating-dominating set in a graph G is a subset of vertices representing "detectors" which can locate an "intruder" given that each detector covers its closed neighborhood and can distinguish its own location from its neighbors. We explore a fault-tolerant variant of locating-dominating sets called redundant locating-dominating sets, which can tolerate one detector malfunctioning (going offline or being removed). In particular, we characterize redundant locating-dominating sets and prove that the problem of determining the minimum cardinality of a redundant locating-dominating set is NP-complete. We also determine tight bounds for the minimum density of redundant locating-dominating sets in several classes of graphs including paths, cycles, ladders, k-ary trees, and the infinite hexagonal and triangular grids. We find tight lower and upper bounds on the size of minimum redundant locating-dominating sets for all trees of order $n$, and characterize the family of trees which achieve these two extremal values, along with polynomial time algorithms to classify a tree as minimum extremal or not.
A locating-dominating set is a subset of vertices representing "detectors" in a graph G; each detector monitors its closed neighborhood and can distinguish its own location from its neighbors, and given all sensor input, the system can locate an "intruder" anywhere in the graph. We explore a fault-tolerant variant of locating-dominating sets, error-correcting locating-dominating (ERR:LD) sets, which can tolerate an incorrect signal from a single detector. In particular, we characterize error-correcting locating-dominating sets, and derive its existence criteria. We also prove that the problem …
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 identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.
A dominating set $S$ of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of $S$, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize classes of trees according to certain relationships between their metric-location-dominatio…
In this article, we study locating-dominating codes in binary Hamming spaces $\mathbb{F}^n$. Locating-dominating codes have been widely studied since their introduction in 1980s by Slater and Rall. They are dominating sets suitable for distinguishing vertices in graphs. Dominating sets as well as locating-dominating codes have been studied in Hamming spaces in multiple articles. Previously, Honkala et al. (2004) have presented a lower bound for locating-dominating codes in binary Hamming spaces. In this article, we improve the lower bound for all values $n\geq10$. In particular, when $n=11$, w…
A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the location-domination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found.
In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das \cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for l…
An open-locating-dominating set (OLD-set) is a subset of vertices of a graph such that every vertex in the graph has at least one neighbor in the set and no two vertices in the graph have the same set of neighbors in the set. This is an analogue to the well-studied identifying code in the literature. In this paper, we prove that the optimal density of the OLD-set for the infinite triangular grid is $4/13$.
Let $G=(V,E)$ be a graph and let $r\ge 1$ be an integer. For a set $D \subseteq V$, define $N_r[x] = \{y \in V: d(x, y) \leq r\}$ and $D_r(x) = N_r[x] \cap D$, where $d(x,y)$ denotes the number of edges in any shortest path between $x$ and $y$. $D$ is known as an $r$-identifying code ($r$-locating-dominating set, respectively), if for all vertices $x\in V$ ($x \in V\backslash D$, respectively), $D_r(x)$ are all nonempty and different. In this paper, we provide complete results for $r$-identifying codes in paths and odd cycles; we also give complete results for 2-locating-dominating sets in cyc…
A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating-dominating. Locating, metric-locating-dominating and locating-dominating sets of minimum cardinality are called b-codes, e-codes and l-codes, respectively. A Nordhaus-Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its…
A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $γ^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin…
A locating-dominating set of a graph $G$ is a dominating set of $G$ such that every vertex of $G$ outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of $G$ is the minimum cardinality of a locating-dominating set in $G$. Let $G_{1}$ and $G_{2}$ be the disjoint copies of a graph $G$ and $f:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F^f_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:v=f(u)\}$. In this paper, we study the variation of the location-…
A note on partitioning the vertex set of a graph into a dominating set and a locating dominating set
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学）；University of Johannesburg（约翰内斯堡大学）；Lebanese American University（黎巴嫩美国大学）；Centrale Méditerranée, Laboratoire d’Informatique et Systèmes UMR 7020（地中海中央理工学院）；University of Turku（图尔库大学）
Univ. of Blida（布利达大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学）；Department of Mathematics, University of Cádiz（加的斯大学）
In this paper, we continue the study of locating-paired-dominating set, abbreviated LPDS, in graphs introduced by McCoy and Henning. Given a finite or infinite graph $G=(V,E)$, a set $S\subset V$ is paired-dominating if the induced subgraph $G[S]$ has a perfect matching and every vertex in $V$ is adjacent to a vertex in $S$. The other condition for LPDS requires that for any distinct vertices $u,v \in V\backslash S$, we have $N(u)\cap S\neq N(v)\cap S$. Motivated by the conjecture of Kinawi, Hussain and Niepel, we prove the minimal density of LPDS in the king grid is between $8/37$ and $2/9$, …
A locating-dominating set (LDS) of a graph $G$ is a dominating set $S$ of $G$ such that for every two vertices $u$ and $v$ in $V(G) \setminus S$, $N(u)\cap S \neq N(v)\cap S$. The locating-domination number $γ^{L}(G)$ is the minimum cardinality of a LDS of $G$. Further if $S$ is a total dominating set then $S$ is called a locating-total dominating set. In this paper we determine the domination, total domination, locating-domination and locating-total domination numbers for hypertrees and sibling trees.
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 $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of $G$, denoted $LT(G)$,…
A set $D$ of vertices of a graph $G$ is locating if every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$, 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 set (respectively, locating-total dominating set) of $G$. A graph $G$ is twin-free if every two distinct vertices of $G$ have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between…
A locating-dominating set of a 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 distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $γ_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. It is conjectured [D. Garijo, A. González and A. Márquez. The diffe…
In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our more significant contributions on λ-codes and η-codes concerning concerning bounds, extremal values and realization theorems.
A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The \emph{location-domination number} of $G$, $λ(G)$, is the minimum cardinality of a locating-dominating set. In this work we study relationships between $λ({G})$ and $λ(\overline{G})$ for bipartite graphs. The main result is the characterization of all connected bipartite graphs $G$ satisfying $λ(\overline{G})=λ({G})+1$. To this aim, we define an edge-labeled graph $G^S$ as…
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 $γ_L(G)$ of a digraph $G$ is the smallest size of a locating-dominating set of $G$. We investigate upper bounds on $γ_L(G)$ 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 t…
Identifying and locating-dominating codes have been studied widely in circulant graphs of type $C_n(1,2,3,\dots, r)$ over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs $C_n(1,d)$ for $d=3$ and proposed as an open question the case of $d > 3$. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs $C_n(1,d)$, $C_n(1,d-1,d)$ and $C_n(1,d-1,d,d+1)$. We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that t…
A locating-dominating set in an undirected graph is a subset of vertices $S$ such that $S$ is dominating and for every $u,v \notin S$, we have $N(u)\cap S\ne N(v)\cap S$. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set $S$ such that for every $w\in V$, $N[w]^-\cap S=\emptyset$ and for each pair of vertices $u,v\in V\setminus S$, $N^-(u)\cap S\ne N^-(v)\cap S$. We consider the following two parameters. Given an undirected graph $G$, we look for $\overset{\rightarrow}γ_{LD}(G)$ ($\overset{\rightarrow}Γ_{LD}(G))$ which is the…
A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words, S is dominating if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are all nonempty and distinct. A set S \subseteq V(G) is called an identifying code in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks…
A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the location-domination number $λ(G)$. An LD-set $S$ of a graph $G$ is global if it is an LD-set of both $G$ and its complement $\overline{G}$. The global location-domination number $λ_g(G)$ is the minimum cardinality of a global LD-set of $G$. In this work, we give some relations between locating-domina…
The $n/2$-bound for locating-dominating sets in subcubic graphs
Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS（克莱蒙奥弗涅大学，法国国家科学研究中心，圣艾蒂安矿业学院，克莱蒙奥弗涅国立理工学院，LIMOS）；Department of Mathematics and Applied Mathematics, University of Johannesburg（约翰内斯堡大学数学与应用数学系）；Department of Computer Science and Mathematics, Lebanese American University（黎巴嫩美国大学计算机科学与数学系）；Aix-Marseille Université, CNRS, École Centrale Méditerranée, LIS（艾克斯-马赛大学，法国国家科学研究中心，地中海中央理工学院，LIS）；Department of Computer Science and Engineering, SRM University-AP（SRM大学AP校区计算机科学与工程系）；Turku Collegium for Science, Medicine and Technology TCSMT, University of Turku（图尔库科学技术医学学院，图尔库大学）；Department of Mathematics and Statistics, University of Turku（图尔库大学数学与统计系）
We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted IDENTIFYING CODE, (OPEN) LOCATING-DOMINATING SET and METRIC DIMENSION) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diame…
Let $G$ be a graph with vertex set $V$, and let $k$ be a positive integer. A set $D \subseteq V$ is a \emph{distance-$k$ dominating set} of $G$ if, for each vertex $u \in V-D$, there exists a vertex $w\in D$ such that $d(u,w) \le k$, where $d(u,w)$ is the minimum number of edges linking $u$ and $w$ in $G$. Let $d_k(x, y)=\min\{d(x,y), k+1\}$. A set $R\subseteq V$ is a \emph{distance-$k$ resolving set} of $G$ if, for any pair of distinct $x,y\in V$, there exists a vertex $z\in R$ such that $d_k(x,z) \neq d_k(y,z)$. The \emph{distance-$k$ domination number} $γ_k(G)$ (\emph{distance-$k$ dimension…
A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $γ(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $γ$-set. A generalization of this is partial domination which was introduced in 2017 by Case, Hedetniemi, Laskar, and Lipman [3,2] . In partial domination a set $S$ is a $p$-dominating set if it dominates a proportion $p$ of the vertices in $V$. The p-domination number $γ_{p}(G)$ is the minimum cardinality of a $p$-dominating set in $G$. In this pap…
A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the \emph{location-domination number} $λ(G)$. An LD-set $S$ of a graph $G$ is \emph{global} if it is an LD-set of both $G$ and its complement $\overline{G}$. The \emph{global location-domination number} $λ_g(G)$ is the minimum cardinality of a global LD-set of $G$. For any LD-set $S$ of a given graph $G$…
For a graph $G$, the $γ$-graph of $G$, $G(γ)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(γ)$ are adjacent if and only if their corresponding dominating sets in $G$ differ by exactly two adjacent vertices. In this paper, we present several variations of the $γ$-graph including those using identifying codes, locating-domination, total-domination, paired-domination, and the upper-domination number. For each, we show that for any graph $H$, there exist infinitely many graphs whose $γ$-graph variant is isomorphic to $H$.
A detection system, modeled in a graph, uses "detectors" on a subset of vertices to uniquely identify an "intruder" at any vertex. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). An OLD set gives each vertex a unique, non-empty open neighborhood of detectors, while an IC provides a unique, non-empty closed neighborhood of detectors. We explore their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector guarantees the properties of OLD sets and ICs,…
This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider sp…
