Locating-dominating Codes Research Articles

On Three Domination-based Identification Problems in Block Graphs

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.

Optimal Local Identifying and Local Locating-dominating Codes

We introduce two new classes of covering codes in graphs for every positive integer r. These new codes are called local r-identifying and local r-locating-dominating codes and they are derived from r-identifying and r-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small n optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid and the king grid. We prove that seven out of eight of our constructions have optimal densities.

Improved lower bound for locating-dominating codes in binary Hamming spaces

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 $n=11$, we manage to improve the previous lower bound from $309$ to $317$. This value is very close to the current best known upper bound of $320$.

Locating and Identifying Codes in Circulant Graphs

Identifying and locating-dominating codes have been studied widely in circulant graphs. Recently, Ville Junnila et al. (Optimal bounds on codes for location in circulant graphs, Cryptography and Communications; 2019) studied identifying and locating-dominating codes in circulants C n 1 , d , C n 1 , d − 1 , d , and C n 1 , d − 1 , d , d + 1 . In this paper, identifying, locating, and self-identifying codes in the circulant graphs C n k , d , C n k , d − k , d , and C n k , d − k , d , d + k are studied, and this extends Junnila et al.’s results to general cases.

Linear-time algorithms for three domination-based separation problems in block graphs

The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view, even for several graph classes where other in general hard problems are easy to solve, like bipartite graphs or chordal graphs. Hence, a typical line of attack for these problems is to determine minimum codes of special graphs. In this work we study the problem of determining the cardinality of minimum such codes in block graphs (that are diamond-free chordal graphs). We present linear-time algorithms for these problems, as a generalization of a linear-time algorithm proposed by Auger in 2010 for identifying codes in trees. Thereby, we provide a subclass of chordal graphs for which all three problems can be solved in linear time.

Exact values for three domination-like problems in circular and infinite grid graphs of small height

In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on a generic framework, that we developed earlier, for the search of a minimum labeling satisfying a pseudo-d-local property in rotagraphs.

Optimal bounds on codes for location in circulant graphs

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 these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in $C_{n}(1,3)$ and $C_{n}(1,4)$ .

On regular and new types of codes for location-domination

Identifying codes and locating-dominating codes have been designed for locating irregularities in sensor networks. In both cases, we can locate only one irregularity and cannot even detect multiple ones. To overcome this issue, self-identifying codes have been introduced which can locate one irregularity and detect multiple ones. In this paper, we define two new classes of locating-dominating codes which have similar properties. These new locating-dominating codes as well as the regular ones are then more closely studied in the rook’s graphs and binary Hamming spaces.In the rook’s graphs, we present optimal codes, i.e., codes with the smallest possible cardinalities, for regular location-domination as well as for the two new classes. In the binary Hamming spaces, we present lower bounds and constructions for the new classes of codes; in some cases, the constructions are optimal. Moreover, one of the obtained lower bounds improves the bound of Honkala et al. (2004) on codes for locating multiple irregularities.Besides studying the new classes of codes, we also present record-breaking constructions for regular locating-dominating codes. In particular, we present a locating-dominating code in the binary Hamming space of length 11 with 320 vertices improving the earlier bound of 352; the best known lower bound for such code is 309 by Honkala et al. (2004).

On the ensemble of optimal dominating and locating-dominating codes in a graph

Let G be a simple, undirected graph with vertex set V. For every v∈V, we denote by N(v) the set of neighbours of v, and let N[v]=N(v)∪{v}. A set C⊆V is said to be a dominating code in G if the sets N[v]∩C, v∈V, are all nonempty. A set C⊆V is said to be a locating-dominating code in G if the sets N[v]∩C, v∈V∖C, are all nonempty and distinct. The smallest size of a dominating (resp., locating-dominating) code in G is denoted by d(G) (resp., ℓ(G)).We study the ensemble of all the different optimal dominating (resp., locating-dominating) codes C, i.e., such that |C|=d(G) (resp., |C|=ℓ(G)) in a graph G, and strongly link this problem to that of induced subgraphs of Johnson graphs.

New results on variants of covering codes in Sierpiński graphs

In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpiński graphs. We compute the minimum size of such codes in Sierpiński graphs.

On $(r,\leq 2)$-locating-dominating codes in the infinite king grid

Assume that $G=(V,E)$ is an undirected graph with vertex set $V$ and edge set $E$.The ball $B_r(v)$ denotes the vertices within graphical distance $r$ from $v$.Let $I_r(F)=\bigcup$$v\in F$$(B_r(v) \cap C)$ be a set of codewords in the neighbourhoods of vertices $v\in F$.A subset $C\subseteq V$ is called an $(r,\leq l)$-locating-dominating code of type A if sets $I_r(F_1)$ and $I_r(F_2)$are distinct for all subsets $F_1,F_2\subseteq V$ where $F_1\ne F_2$, $F_1\cap C= F_2 \cap C$ and $|F_1|,|F_2| \leq l$.A subset $C\subseteq V$ is an $(r,\leq l)$-locating-dominating code of type B if the sets $I_r(F)$ are distinct for all subsets$F\subseteq V\setminus C$ with at most $l$ vertices. We study $(r,\leq l)$-locating-dominating codes in the infinite king gridwhen $r\geq 1$ and $l=2$. The infinite king grid is the graph with vertex set $\mathbb Z^2$ and edge set$\{\{(x_1,y_1),(x_2,y_2)\}||x_1-x_2|\leq 1, |y_1-y_2|\leq 1, (x_1,y_1)\ne(x_2,y_2)\}.$

Extremal cardinalities for identifying and locating-dominating codes in graphs

Consider a connected undirected graph G = ( V , E ) , a subset of vertices C ⊆ V , and an integer r ⩾ 1 ; for any vertex v ∈ V , let B r ( v ) denote the ball of radius r centred at v , i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V (respectively, v ∈ V ⧹ C ), the sets B r ( v ) ∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the extremal values of the cardinality of a minimum r-identifying or r-locating-dominating code in any connected undirected graph G having a given number, n, of vertices. It is known that a minimum r-identifying code contains at least ⌈ log 2 ( n + 1 ) ⌉ vertices; we establish in particular that such a code contains at most n - 1 vertices, and we prove that these two bounds are reached. The same type of results are given for locating-dominating codes.

Identifying codes of cycles

In this paper we deal with identifying codes in cycles. We show that for all r≥1, any r-identifying code of the cycle Cn has cardinality at least gcd(2r+1,n)⌈n2gcd(2r+1,n)⌉. This lower bound is enough to solve the case n even (which was already solved in [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (7) (2004) 969–987]), but the case n odd seems to be more complicated. An upper bound is given for the case n odd, and some special cases are solved. Furthermore, we give some conditions on n and r to attain the lower bound.

Identifying and locating-dominating codes on chains and cycles

Consider a connected undirected graph G=( V, E), a subset of vertices C⊆ V, and an integer r≥1; for any vertex v∈ V, let B r ( v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v∈ V (respectively, v∈ V ⧹ C), the sets B r ( v)∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles.

On Locating-Dominating Codes in Binary Hamming Spaces

Locating faulty processors in a multiprocessor system gives the motivation for locating-dominating codes. We consider these codes in binary hypercubes and generalize the concept for the situation in which we want to locate more than one malfunctioning processor.

Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Let G=(V,E) be an undirected graph and C a subset of vertices. If the sets Br(v)∩C, v∈V (respectively, v∈V⧹C), are all nonempty and different, where Br(v) denotes the set of all points within distance r from v, we call C an r-identifying code (respectively, an r-locating-dominating code). We prove that, given a graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r.

Identifying and locating-dominating codes: NP-completeness results for directed graphs

Let G=(V, A) be a directed, asymmetric graph and C a subset of vertices, and let B/sub r//sup -/(v) denote the set of all vertices x such that there exists a directed path from x to v with at most r arcs. If the sets B/sub r//sup -/(v) /spl cap/ C, v /spl isin/ V (respectively, v /spl isin/ V/spl bsol/C), are all nonempty and different, we call C an r-identifying code (respectively, an r-locating-dominating code) of G. In other words, if C is an r-identifying code, then one can uniquely identify a vertex v /spl isin/ V only by knowing which codewords belong to B/sub r//sup -/(v), and if C is r-locating-dominating, the same is true for the vertices v in V/spl bsol/C. We prove that, given a directed, asymmetric graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r/spl ges/1 and remains so even when restricted to strongly connected, directed, asymmetric, bipartite graphs or to directed, asymmetric, bipartite graphs without directed cycles.