On two variations of identifying codes

Identifying codes have been introduced in 1998 to model fault-detection in multiprocessor systems. In this paper, we introduce two variations of identifying codes: weak codes and light codes. They correspond to fault-detection by successive rounds. We give exact bounds for those two definitions for the family of cycles

Identifying codes and locating–dominating sets on paths and cycles

AbstractLet G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define Nr[x]={y∈V:d(x,y)≤r} and Dr(x)=Nr[x]∩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∈V (x∈V∖D, respectively), Dr(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72–82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating–dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969–987]

Optimal Identifying Codes in Cycles and Paths

The concept of identifying codes in a graph was introduced by Karpovsky et al. (in IEEE Trans Inf Theory 44(2):599-611, 1998). These codes have been studied in several types of graphs such as hypercubes, trees, the square grid, the triangular grid, cycles and paths. In this paper, we determine the optimal cardinalities of identifying codes in cycles and paths in the remaining open cases

Solving Two Conjectures regarding Codes for Location in Circulant Graphs

Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs

Polyhedra associated with identifying codes

In this work we study the associated polyhedra and present some general results on their combinatorial structure. We demonstrate how the polyhedral approach can be applied to find minimum identifying codes for special bipartite graphs and cycles, and discuss further lines of research in order to obtain strong lower bounds stemming from linear relaxations of the identifying code polyhedron, enhanced by suitable cutting planes to be used in a B&C framework.Sociedad Argentina de Informática e Investigación Operativ

Polyhedra associated with identifying codes

In this work we study the associated polyhedra and present some general results on their combinatorial structure. We demonstrate how the polyhedral approach can be applied to find minimum identifying codes for special bipartite graphs and cycles, and discuss further lines of research in order to obtain strong lower bounds stemming from linear relaxations of the identifying code polyhedron, enhanced by suitable cutting planes to be used in a B&C framework.Sociedad Argentina de Informática e Investigación Operativ

Adaptive Identification of Sets of Vertices in Graphs

We present an optimal adaptive algorithm for identifying vertices in cycles. We also give efficient adaptive algorithms for identifying sets of vertices in different graphs such as cycles, king lattices and square lattices. Adaptive identification is also considered in Hamming spaces, which is one of the most widely studied graphs in the field of identifying codes

Identifying and locating-dominating codes on chains and cycles

AbstractConsider a connected undirected graph G=(V,E), a subset of vertices C⊆V, and an integer r≥1; for any vertex v∈V, let Br(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 Br(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