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).</p

Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem

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

Random subgraphs make identification affordable

An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every graph $G$ with $n$ vertices, maximum degree $\Delta=\omega(1)$ and minimum degree $\delta\geq c\log{\Delta}$, for some constant $c>0$, contains a large spanning subgraph which admits an identifying code with size $O\left(\frac{n\log{\Delta}}{\delta}\right)$. In particular, if $\delta=\Theta(n)$, then $G$ has a dense spanning subgraph with identifying code $O\left(\log n\right)$, namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code

Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates

We construct an exactly solvable Hamiltonian acting on a 3-dimensional lattice of spin-$\frac 1 2$ systems that exhibits topological quantum order. The ground state is a string-net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends upon the homology of the 3-manifold. We generalize the system to $D\geq 4$, were different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.Comment: Revtex4 file, color figures, minor correction

Automated Discharging Arguments for Density Problems in Grids

Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of $\frac{3}{7} \approx 0.428571$ and a lower bound of $\frac{5}{12}\approx 0.416666$. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of $\frac{23}{55} \approx 0.418181$ on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables, and 2 figure