On the size of identifying codes in triangle-free graphs

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let \M(G) be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta\geq 3$, \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c holds for any nontrivial connected identifiable graph $G$

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

Transversal Diagonal Logical Operators for Stabiliser Codes

Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.Comment: 24 pages + 11 page appendix, 4 figures, comments welcom

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

Identifying codes of corona product graphs

For a vertex $x$ of a graph $G$, let $N_G[x]$ be the set of $x$ with all of its neighbors in $G$. A set $C$ of vertices is an {\em identifying code} of $G$ if the sets $N_G[x]\cap C$ are nonempty and distinct for all vertices $x$. If $G$ admits an identifying code, we say that $G$ is identifiable and denote by $\gamma^{ID}(G)$ the minimum cardinality of an identifying code of $G$. In this paper, we study the identifying code of the corona product $H\odot G$ of graphs $H$ and $G$. We first give a necessary and sufficient condition for the identifiable corona product $H\odot G$, and then express $\gamma^{ID}(H\odot G)$ in terms of $\gamma^{ID}(G)$ and the (total) domination number of $H$. Finally, we compute $\gamma^{ID}(H\odot G)$ for some special graphs $G$

Open-independent, Open-locating-dominating Sets

A distinguishing set for a graph G = (V, E) is a dominating set D, each vertex $v \in D$ being the location of some form of a locating device, from which one can detect and precisely identify any given "intruder" vertex in V(G). As with many applications of dominating sets, the set $D$ might be required to have a certain property for <D>, the subgraph induced by D (such as independence, paired, or connected). Recently the study of independent locating-dominating sets and independent identifying codes was initiated. Here we introduce the property of open-independence for open-locating-dominating sets