The metric dimension for resolving several objects

A set of vertices S is a resolving set in a graph if each vertex has a unique array of distances to the vertices of S. The natural problem of finding the smallest cardinality of a resolving set in a graph has been widely studied over the years. In this paper, we wish to resolve a set of vertices (up to l vertices) instead of just one vertex with the aid of the array of distances. The smallest cardinality of a set S resolving at most l vertices is called l-set-metric dimension. We study the problem of the l-set-metric dimension in two infinite classes of graphs, namely, the two dimensional grid graphs and the n-dimensional binary hypercubes. (C) 2016 Elsevier B.V. All rights reserved

Minimum Number of Input Clues in Robust Information Retrieval

Information retrieval in associative memories was considered recently by Yaakobi and Bruck. In their model, a stored information unit is retrieved using input clues. In this paper, we study the problem where at most s (s >= 0) of the received input clues can be false and we still want to determine the sought information unit uniquely. We use a coding theoretical approach to estimate the maximum number of stored information units with respect to a given s. Moreover, optimal results for the problem are given, for example, in the infinite king grid. We also discuss the problem in the class of line graphs where a characterization and a connection to k-factors is given

Identification in Z(2) using Euclidean balls

The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin. These codes find their application, for example, in sensor networks. The network is modelled by a graph. In this paper, the goal is to find good identifying codes in a natural setting, that is, in a graph epsilon(r) = (V, E) where V = Z(2) is the set of vertices and each vertex (sensor) can check its neighbours within Euclidean distance r. We also consider a graph closely connected to a well-studied king grid, which provides optimal identifying codes for epsilon(root 5) and epsilon(root 13). (C) 2010 Elsevier B.V. All rights reserved

Information Retrieval With Varying Number of Input Clues

Information retrieval in associative memories was studied in a recent paper by Yaakobi and Bruck (2012). Associations between memory entries give us the t-neighbourhood of an entry. In their model, an information unit is retrieved from the memory with the aid of input clues, which are chosen from a reference set. In this paper, we consider the situation where the information unit is found unambiguously using the associated t-neighbourhoods of the input clues. A varying number of input clues are allowed, but a limit m(u) on the maximum number of them is imposed. Of course, we would like m(u) to be as small as possible. We consider the problem over the binary Hamming space F-n and focus on the minimum of m(u), denoted by.(n; t). Using linear reference sets, we show that.(n; 2) = 9. We also give infinite families of reference sets, which provide good bounds on.(n; t) for t = 3. In addition, efficient methods are given to obtain bounds on.(n; t) for any t from known reference sets. We also discuss the applications of this model to the Levenshtein's sequence reconstruction problem and the sensor network monitoring.</p

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

Locating-dominating codes in cycles

The smallest cardinality of an r-locating-dominating code in a cycle C_n of length n is denoted by M_r^{LD}(C_n). In this paper, we prove that for any r geq 5 and n geq n_r when n_r is large enough (n_r=mathcal{O}(r^3)) we have n/3 leq M_r^{LD}(C_n) leq n/3+1 if n equiv 3 pmod{6} and M_r^{LD}(C_n) = lceil n/3 ceil otherwise. Moreover, we determine the exact values of M_3^{LD}(C_n) and M_4^{LD}(C_n) for all n

Solving Two Conjectures regarding Codes for Location in Circulant Graphs

Identifying and locating-dominating codes have been widely studied in circulant graphs of type Cn(1, 2, . . ., 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 Cn(1, 3). They showed that the smallest cardinality of a locating-dominating code in Cn(1, 3) is at least ⌈n/3⌉ and at most ⌈n/3⌉ + 1 for all n ≥ 9. Moreover, they proved that the lower bound is strict when n ≡ 0, 1, 4 (mod 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 Cn(1, 3) is at least ⌈4n/11⌉ and at most ⌈4n/11⌉ + 1 for all n ≥ 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.</p

Locating-dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1] r-locating-dominating codes in paths P(n). They conjectured that if r >= 2 is a fixed integer, then the smallest cardinality of an r-locating-dominating code in P(n), denoted by M(r)(LD) (P(n)), satisfies M(r)(LD)(P(n)) = [(n + 1)/3] for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r >= 3 we have M(r)(LD) (P(n)) = [(n + 1)/3] for all n >= n(r), when n(r) is large enough. In addition, we solve a conjecture on location-domination with segments of even length in the infinite path. (C) 2011 Elsevier B.V. All rights reserved

Insect oviposition preference between Epichloe-symbiotic and Epichloe-free grasses does not necessarily reflect larval performance

Variation in plant communities is likely to modulate the feeding and oviposition behavior of herbivorous insects, and plant-associated microbes are largely ignored in this context. Here, we take into account that insects feeding on grasses commonly encounter systemic and vertically transmitted (via seeds) fungal Epichloe endophytes, which are regarded as defensive grass mutualists. Defensive mutualism is primarily attributable to alkaloids of fungal origin. To study the effects of Epichloe on insect behavior and performance, we selected wild tall fescue (Festuca arundinacea) and red fescue (Festuca rubra) as grass-endophyte models. The plants used either harbored the systemic endophyte (E+) or were endophyte-free (E-). As a model herbivore, we selected the Coenonympha hero butterfly feeding on grasses as larvae. We examined both oviposition and feeding preferences of the herbivore as well as larval performance in relation to the presence of Epichloe endophytes in the plants. Our findings did not clearly support the female's oviposition preference to reflect the performance of her offspring. First, the preference responses depended greatly on the grass-endophyte symbiotum. In F. arundinacea, C. hero females preferred E+ individuals in oviposition-choice tests, whereas in F. rubra, the endophytes may decrease exploitation, as both C. hero adults and larvae preferred E- grasses. Second, the endophytes had no effect on larval performance. Overall, F. arundinacea was an inferior host for C. hero larvae. However, the attraction of C. hero females to E+ may not be maladaptive if these plants constitute a favorable oviposition substrate for reasons other than the plants' nutritional quality. For example, rougher surface of E+ plant may physically facilitate the attachment of eggs, or the plants offer greater protection from natural enemies. Our results highlight the importance of considering the preference of herbivorous insects in studies involving the endophyte-symbiotic grasses as host plants