The watching system as a generalization of identifying code

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Abstract

The watching system, as a generalization of identifying code, has been defined by Auger in 2010. The identifying code has been used to wireless networks and it has been also applied to locate objects in the sensor networks. On the other hand, the graph product is employed in most of the mathematic branches such as network design to study the structure of network elements. In this paper, we give some upper bounds for the watching number of well-know product graphs.

Introduction

In this text, we consider only the simple, connected and undirected graphs. For the vertex v of G, let the subset CV(G) be such that IC(v)=CNG[v]. If IC(v), for all vertices of G, is non-empty and distinct, then C is called an identifying code. When the set C exists, the minimum sizes of C is called the identifying code denoted by i(G).

The identifying codes have been developed to explore internal detection systems in wireless networks or to model fault diagnosis in multiprocessor systems. Identifying codes have been widely studied in both graph theory and coding theory, see [4], [6], [7]. Auger et al. in [1] generalized the definition of identifying codes to watching systems.

Many mathematical or real-life problems come down to investigate whether a particular element lies in a given set X of possible places. For example, a multiprocessor system can be modeled as a graph, where the set of processors are the set of vertices and the set of links in the system composes the edge set. Some software carries out the diagnosis of selected processors. These processors can be selected by generating the code that allows for the unique identification of faulty processors. A processor corresponding to a codeword can test itself and all its neighbors. In other words, in a multiprocessor system, a codeword is a processor equipped with a sensor, with the ability to detect a faulty processor if it lies in its closed neighborhood. If the number of faulty processors is at most one and if each codeword transfer only one bit of information, whether it detects a faulty processor or not, then an identifying tell us to investigate, from the |C| bits of information received, whether there is a fault in the graph or not. Consider the graph G as depicted in Figure 1, we need five codewords (this is equal with n1, the maximum number of watchers required for a graph on n vertices to monitor all vertices) to watch all six vertices. It is necessary to check the whole closed neighbourhood so that two distinct codewords check almost the same sets of vertices. Let us use the term watcher instead of codeword for this generalization. This yields that for example, a watching system can models a fire-monitoring system in a building. Based the above explanation, a watching system can be defined as follows:

We say a couple ω=(υi,Zi) is a watcher ω in G, where υi is a vertex, ZiNG[υi] and vi is covered by ω and Zi is the area of watching and ω is called a watcher. The vertex vi is called a watched vertex and a vertex not covered by a watcher is called a non-watched vertex.

For given graph G, a finite set W={ω1,ω2,,ωk} is a watching system, where ωi, 1 ≤ i ≤ k is a watcher such that {Z1,Z2,,Zk} is an identifying system. So, W is a watching system for G, if all vertices have non-empty and different labels. We show the minimum number of watchers needed to cover all vertices of G, by ω(G). For example, in Figure 2, we have ω(G)=3.

Hence, in a watching system, the selection of neighbor vertices is arbitrary as a watching area from a watcher. This is the difference between a watching system and an identifying code.

Section snippets

Main Results

The aim of this section is to obtain some bounds of watching number of product graphs. The relationship between watching number and the size of the vertex set of G is expressed in [1], [2]. In other words, for a graph G, one can see thatlog2(|V(G)|+1)ω(G)andγ(G)ω(G)γ(G)log2(Δ(G)+2),where Δ(G) denotes to the maximum degree of graph G.

Let G be a graph on n vertices. Then the vertex x ∈ V(G) is called a well-connected vertex or a universal vertex, if the degree of x is n1.

Lemma 2.1

[1] If G has at

Acknowledgement

Matthias Dehmer thanks the Austrian Science Fund for supporting this work (P 30031).

References (7)

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