Figure 3 - uploaded by Sylvain Gravier
Content may be subject to copyright.
Source publication
In a graph G=(V,E), an identifying code of G (resp. a locating-dominating code of G) is a subset of vertices C⊆V such that N[v]∩C≠∅ for all v∈V, and N[u]∩C≠N[v]∩C for all u≠v,u,v∈V (resp. u,v∈V [integerdivide] C), where N[u] denotes the closed neighbourhood of v, that is N[u]=N(u)∪{u}. These codes model fault-detection problems in multiprocessor sy...
Similar publications
In wireless sensor network (WSN), the energy of sensor nodes is limited. Designing efficient routing method for reducing energy consumption and extending the WSN’s lifetime is important. This paper proposes a novel energy-efficient, static scenario-oriented routing method of WSN based on edge computing named the NEER, in which WSN is divided into s...
The wireless sensor networks (WSN) are formed by a large number of sensor nodes working together to pro-vide a specific duty. However, the low energy capacity assigned to each node prompts users to look at an important design challenge such as lifetime maximization. Therefore, designing effective routing techniques that conserve scarce energy resou...
In this paper, we propose two routing protocols for Terrestrial Wireless Sensor Networks (TWSNs); Hybrid Energy Efficient Reactive (HEER), and Multi-hop Hybrid Energy Efficient Reactive (MHEER) routing protocol. The main purpose of designing these protocols is to improve the network lifetime and particularly the stability period of the underlying n...
Reliability and energy efficiency are two key considerations when designing a compressive sensing (CS)-based data-gathering scheme. Most researchers assume there is no packets loss, thus, they focus only on reducing the energy consumption in wireless sensor networks (WSNs) while setting reliability concerns aside. To balance the performance–energy...
Designing and reasoning about complex systems such as wireless sensor networks is hard due to highly dynamic environments: sensors are heterogeneous, battery-powered, and mobile. While formal modelling can provide rigorous mechanisms for design/reasoning, they are often viewed as difficult to use. Graph rewrite-based modelling techniques increase u...
Citations
... In particular, generalizing a result from [1], it is shown in [15] that for every graph G of order n and girth at least 5, we have γ ID (G) 5n+2 (G) 7 , a bound which is tight. Relations between identification-type graph parameters were provided in [20] (locatingdominating sets and identifying codes) and [38] (locating-dominating sets, identifying codes, and open-locating-dominating sets). It is shown that for every graph G, any two of these parameters' values cannot be more than a factor 2 apart from each other. ...
... Tight bounds relating the parameters γ ID , γ L and γ OL were provided in the literature. It was indeed proved in [20] that for any identifiable graph G, γ ID (G) 2γ L (G) holds (and is tight). Similar bounds were proved in the PhD thesis [38,Chapter 2.4.1], ...
... Let C LD be an optimal locating-dominating set with at least two codewords in an identifiable connected graph G. We have γ ID (G) 2|C LD | by [20,Theorem 8]. Moreover, following the proof of [20], we may construct an identifying code from C LD by just adding at most |C LD | additional vertices to C LD . ...
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 $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$. Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ 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 upper bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins). Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the similar parameter $\gamma^{\text{ID}}(G)$ as well as to the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.
... Proof. Let A be a polynomial-time (2 ln n + 1)-approximation algorithm for the Separation problem [18]. For any graph G, let S(G) denote the separating set returned by A on the input graph G. Using Theorem 10, we have ...
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 of the two vertices of the pair. We study the computational complexity of Red-Blue Separation, in which one asks whether a given red-blue colored graph has a red-blue separating set of size at most a given integer. We prove that the problem is NP-complete even for restricted graph classes. We also show that it is always approximable in polynomial time within a factor of $2\ln n$, where $n$ is the input graph's order. In contrast, for triangle-free graphs and for graphs of bounded maximum degree, we show that Red-Blue Separation is solvable in polynomial time when the size of the smaller color class is bounded by a constant. However, on general graphs, we show that the problem is $W[2]$-hard even when parameterized by the solution size plus the size of the smaller color class. We also consider the problem Max Red-Blue Separation where the coloring is not part of the input. Here, given an input graph $G$, we want to determine the smallest integer $k$ such that, for every possible red-blue coloring of $G$, there is a red-blue separating set of size at most $k$. We derive tight bounds on the cardinality of an optimal solution of Max Red-Blue Separation, showing that it can range from logarithmic in the graph order, up to the order minus one. We also give bounds with respect to related parameters. For trees however we prove an upper bound of two-thirds the order. We then show that Max Red-Blue Separation is NP-hard, even for graphs of bounded maximum degree, but can be approximated in polynomial time within a factor of $O(\ln^2 n)$.
... In particular, generalizing a result from [1], it is shown in [15] that for every graph G of order n and girth at least 5, we have γ ID (G) ≤ 5n+2ℓ(G) 7 , a bound which is tight. Relations between identification-type graph parameters were provided in [20] (locating-dominating sets and identifying codes) and [38] (locating-dominating sets, identifying codes, and open-locatingdominating sets). It is shown that for every graph G, any two of these parameters' values cannot be more than a factor 2 apart from each other. ...
... Tight bounds relating the parameters γ ID , γ L and γ OL were provided in the literature. It was indeed proved in [20] that for any identifiable graph G, γ ID (G) ≤ 2γ L (G) holds (and is tight). Similar bounds were proved in the PhD thesis [38,Chapter 2.4.1], ...
... Let C LD be an optimal locating-dominating set with at least two codewords in an identifiable connected graph G. We have γ ID (G) ≤ 2|C LD | by [20,Theorem 8]. Moreover, following the proof of [20], we may construct an identifying code from C LD by just adding at most |C LD | additional vertices to C LD . ...
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 $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$. Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ 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 $\gamma_t^{\text{ID}}(G)$ to the related parameter $\gamma^{\text{ID}}(G)$ as well as the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.
... We use the reduction of Gravier et al. [18,Figure 7]. Consider an instance (G, k) of Dominating-Set. ...
... Let G △ be the graph obtained by adding to each vertex of the graph a pendant triangle (see Figure 1). Then it is proved in [18] that G has a dominating set of size k if and only if G △ has a locating-dominating set of size k + n (where n is the number of vertices of G). Indeed, each triangle must contain at least one of the new vertices in a locating-dominating set and if there is exactly one vertex in a triangle, the vertex of the original graph must be dominated in the original graph. ...
A locating-dominating set of an undirected graph is a subset of vertices $S$ such that $S$ is dominating and for every $u,v \notin S$, the neighbourhood of $u$ and $v$ on $S$ are distinct (i.e. $N(u) \cap S \ne N(v) \cap S$). In this paper, we consider the oriented version of the problem. A locating-dominating set of an oriented graph is a set $S$ such that for every $u,v \in V \setminus S$, $N^-(u) \cap S \ne N^-(v) \cap S$. We consider the following two parameters. Given an undirected graph $G$, we look for $\overset{\rightarrow}{\gamma}_{LD}(G)$ ($\overset{\rightarrow}{\Gamma}_{LD}(G))$ which is the size of the smallest (largest) optimal locating-dominating set over all orientations of $G$. In particular, if $D$ is an orientation of $G$, then $\overset{\rightarrow}{\gamma}_{LD}(G)\leq{\gamma}_{LD}(D)\leq\overset{\rightarrow}{\Gamma}_{LD}(G)$. For the best orientation, we prove that, for every twin-free graph $G$ on $n$ vertices, $\overset{\rightarrow}{\gamma}_{LD}(G) \le n/2$ proving a ``directed version'' of a conjecture on $\gamma_{LD}(G)$. Moreover, we give some bounds for $\overset{\rightarrow}{\gamma}_{LD}(G)$ on many graph classes and drastically improve value $n/2$ for (almost) $d$-regular graphs by showing that $\overset{\rightarrow}{\gamma}_{LD}(G) \in O(\log d / d \cdot n)$ using a probabilistic argument. While $\overset{\rightarrow}{\gamma}_{LD}(G)\leq\gamma_{LD}(G)$ holds for every graph $G$, we give some graph classes graphs for which $\overset{\rightarrow}{\Gamma}_{LD}(G)\geq{\gamma}_{LD}(G)$ and some for which $\overset{\rightarrow}{\Gamma}_{LD}(G)\leq {\gamma}_{LD}(G)$. We also give general bounds for $\overset{\rightarrow}{\Gamma}_{LD}(G)$. Finally, we show that for many graph classes $\overset{\rightarrow}{\Gamma}_{LD}(G)$ is polynomial on $n$ but we leave open the question whether there exist graphs with $\overset{\rightarrow}{\Gamma}_{LD}(G)\in O(\log n)$.
... al. in [19] generalized the concept of Identifying Codes, to incorporate robustness properties to deal with faults in sensor networks. Suomela and Gravier, in [32], [33] designed an approximation algorithm for the computation of the Minimum Identifying Code Set (MICS), and provided a log(n)−approximation algorithm. ...
The past couple of decades has witnessed an unprecedented rise in organized crime. This rise, coupled with increasing intricacies of organized crime, poses significant and evolving challenges for international law enforcement authorities. With the passage of time, authorities such as Interpol, have discovered that modern criminal organizations have adopted a networked structure, a shift away from the traditional hierarchical structure. Fluid networked structures make it difficult for the authorities to apprehend individuals associated with each network, and consequently, to disrupt the operations of the network. Various research groups have analyzed prison/courtroom transcripts, to create an organizational structure of known individuals, or a social network of individuals, suspected to be a part of a major drug/terrorist organization. These social networks have been studied fairly extensively from network centrality perspectives, to understand the role of suspect individuals in the network. Additionally, with drug and terror offenses increasing globally, the list of suspect individuals has also been growing over the past decade. As it takes significant amount of technical and human resources to monitor a suspect, an increasing list entails higher resource requirements on the part of the authorities, and monitoring all the suspects soon becomes an impossible task. In this paper, we primarily focus on two types of networks – (i) Drug Trafficking Organizations, and (ii) Terrorist Organizations, and present a methodology for the surveillance of individuals associated with these networks. Our methodology is based on the mathematical notion of Identifying Codes, which ensures reduction in resources on the part of law enforcement authorities, without compromising the ability to uniquely identify a suspect, when they become “active” in drug/terror related activities. Furthermore, we show that our approach requires far lesser resources when compared to strategies adopting standard network centrality measures for the unique identification of individuals. In other words, we show that the strategy of monitoring individuals in such networks, by utilizing centrality measures is wasteful, on the part of the authorities. Finally, we evaluate the efficacy of our approach on real world datasets.
... It in fact remains NP-hard for many subclasses of graphs [4,5]. Furthermore, approximating ( ) is not easy as shown in [6,7,8]. For some variants of domination in graphs, the reader may refer to [9,10,11,12,13,14,15,16,17,18]. ...
A subset í µí± of í µí±(í µí°º) is a dominating set of í µí°º if for every í µí±£ ∈ í µí±(í µí°º)\í µí±, there exists í µí±¥ ∈ í µí± such that í µí±¥í µí±£ ∈ í µí°¸(í µí°º). An identifying code of a graph í µí°º is a dominating set í µí° ¶ ⊆ í µí±(í µí°º) such that for every í µí±£ ∈ í µí±(í µí°º), í µí± í µí°º [í µí±£] ∩ í µí° ¶ is distinct. In this paper, we investigate the identifying code of some special graphs and give some important results. Mathematics Subject Classification: 05C69
... (b) [11] If G is a 1-twin-free graph, then id 1 (G) ≤ 2ld 1 (G). ...
Let G = (V,E) be a finite graph and r ≥ 1 be an integer. For v ϵ V, let B r (v) = {x ϵ V: d(v, x) ≤ r} be the ball of radius r centered at v. A set C ⊆ V is an r-dominating code if for all v ϵ V, we have B r (v) ∩ C ≠ θ; it is an r-locating-dominating code if for all v ϵ V, we have B r (v) ∩ C ≠ θ, and for any two distinct non-codewords x ϵ V \ C, y ϵ V \ C, we have B r (x) ∩ C ≠ B r (y) ∩ C; it is an r-identifying code if for all v ϵ V, we have B r (v) ∩ C ≠ θ, and for any two distinct vertices x ϵ V, y ϵ V, we have B r (x) ∩ C ≠ B r (y) ∩ C. We denote by γ r (G) (respectively, ld r (G) and id r (G)) the smallest possible cardinality of an r-dominating code (respec- tively, an r-locating-dominating code and an r-identifying code). We study how small and how large the three differences id r (G)-ld r (G), id r (G)-γ r (G) and ld r (G) - γ r (G) can be.
... Proposition 4 (Gravier, Klasing and Moncel [17]). Let G be a twin-free graph. ...
We consider the problem of computing identifying codes of graphs and its
fractional relaxation. The ratio between the size of optimal integer and
fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of
vertices of the graph. We focus on vertex-transitive graphs for which we can
compute the exact fractional solution. There are known examples of
vertex-transitive graphs that reach both bounds. We exhibit infinite families
of vertex-transitive graphs with integer and fractional identifying codes of
order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles
(strongly regular graphs based on finite geometries). They also provide
examples for metric dimension of graphs.
... For the converse, the reasoning is similar. This also implies that if Minimum Locating-Dominating Set is NP-hard to α-approximate for graphs of diameter 2 for some α ≥ 2, then Centroidal Dimension is NP-hard to approximate within factor α 2 OP T OP T +1 = α(1−o(1)) 2 for graphs of diameter 2. The positive approximation bound follows, as Minimum Locating-Dominating Set is wellknown to be O(ln n)-approximable, see for example Gravier, Klasing and Moncel [8]. ...
... Moreover, it follows from a reduction for Minimum Identifying Code in the first author's thesis [5, Section 6.4] and a lemma from Gravier, Klasing and Moncel [8] (see also Foucaud [6,7]) that Minimum Locating-Dominating Set is NP-hard to approximate within a factor of o(ln n) for graphs having a vertex adjacent to all other vertices. This proves the non-approximability bound. ...
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, 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, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. 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,
$CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
$CD(G)$ for an input graph $G$, showing that the problem is hard and cannot
even be approximated efficiently up to a factor of $o(\log n)$. We also give an
$O\left(\sqrt{n\ln n}\right)$-approximation algorithm.
... We refer to [45] for an on-line bibliography on these topics, which lists more than 240 papers as of February 2013. In particular, see [2,3,6,17,28,30,31,34,43,49,54,55,56] for studies of the computational complexity of these problems. We remark that in many of these papers, due to the similarity between the two problems, the algorithmic properties of identifying codes and locating-dominating sets are studied together. ...
... We now show a way to relate (non-)approximability results of Min Id Code and Min Loc-Dom Set. The following theorem was given in [34]. ...
... This result holds even for bipartite graphs [17], and for Id Code it holds for planar graphs of maximum degree 3 [2,3], planar bipartite unit disk graphs [49], line graphs [30], split graphs [28,31], and, interestingly, interval graphs [28,31]. Regarding the minimization problems, log-APX-completeness of Min Id Code and Min Loc-Dom Set is known only for general graphs [6,43,56], and the two problems are APX-complete for graphs of bounded maximum degree at least 8 and 5, respectively [34]. ...
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 of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum Locating-Dominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.
