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De nition 9. Graph Weight. The nodes in the network graph can have a positive weight wv . The total weight of a cluster is given by Wsum (C) =
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Dominating Sets and Covers. Another group of graph de nitions which are very useful when trying to model the clusters is the concept of sets. Set de nition in the context of WSN are listed below. By being able of de ning these sets in the network, the nodes can calculate their real coverage and establish the stability of the communication paths between them. Generally, the clusters are de ned using the vertex cover of the graph (see De nition 10) and the nodes that belong to that vertex cover are selected as the cluster heads. The remaining nodes should calculate the stability of their communication link to the neighboring CHs and join the cluster corresponding to the link with the best connection stability. De nition 10. Vertex Cover. A vertex cover is a subset S V , such that for all edges in E, S e = 0, which means that every edge has at least one edge in S. / De nition 11. Independent Set. The independent set ISG is the subset S V , such that there is no edge between any pair of nodes in S. The maximum independent set in G is equal to |V |, the size of the minimum vertex cover of G. De nition 12. Dominating Set. The dominating set DSG is the subset S V such that every node is in S or if it is in V S and has at least one neighbor (adjacent vertex) in S. A vertex of S is said to dominate itself and all adjacent vertices. An edge is dominated if either of its endpoints is in S. The nondominated edges are called free edges. De nition 13. Minimum Dominating Set. The Minimum Dominating Set (MDS ) problem is the problem of nding a dominating set of minimum size. This is an NP-complete problem. De nition 14. Independent Dominating Set. The independent dominating set IDS G is the set S V that is both dominating and independent. De nition 15. Connected Dominating Set. The connected dominating set CDS G is a dominating set whose induced subgraph S is connected. De nition 16. Weakly Induced Subgraph. For any subset S V , the subgraph weakly induced by S, Sw , is the graph (N[S], E (N[S] S). This means that Sw contains the vertices of S, their neighbors, and all edges with at least one endpoint in S. A vertex subset S is a weakly connected dominating set if S is dominating and Sw is connected.
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7.3 GRAPH-BASED APPROACHES FOR CLUSTERING IN WSN In this section, we present a classi cation of proposed clustering techniques based on the application of Graph Theory concepts. Although many of the references are related with general wireless and ad hoc networks and not speci cally WSN, we include them either because such approaches can be applied to WSN, especially static WSN, or because we need to clarify why they are not suitable for WSN. We summarize the main characteristics for each one of the algorithms and protocols, with emphasis on describing the ones suitable for WSN. Once it has been decided that the creation of clusters inside the WSN is an appropriate solution to support other network and application functions, it is necessary to clearly de ne the characteristics desired in the clusters to obtain a well-formed structure. Some of those characteristics are as follows: 1. Every node should be in exactly one cluster. The objective of this is to maximize the average cluster sizes while maintaining full coverage. 2. Guarantee the total coverage of the network. 3. Minimize the number of CH to provide an ef cient network coverage while minimizing the cluster overlap. A minimum cluster overlap reduces the amount of channel contention between clusters and improves the ef ciency of algorithms that execute at the level of the CH. 4. Create a highly uniform, balanced clustering. As an example of the importance of highly uniform clustering with low overlap, consider the clustered broadcast protocol described by Ni et al. [12]. In this protocol, the broadcast message is relayed from CH to CH, which then broadcast the message to their associated nodes in each cluster, called followers. In a clustering with few CH and large cluster sizes, the clusters have minimal overlap and provide the best coverage of the network with the fewest clusters. In this con guration, the number of repeated broadcast transmissions over any area will be small, thus reducing the amount of transmission collisions and channel contention. This allows for faster, more ef cient and more reliable communications. On the other hand, a poor clustering containing a lot of cluster overlap and a large number of CHs lose most of the bene ts of clustering because transmissions will be repeated in areas of overlap with signi cant channel contention In the following subsections we make a classi cation of the graph theory based clustering protocols for static and mobile WSN. This classi cation contains two main groups as shown in Figure 7.3: static-based WSN algorithms and mobile-based WSN algorithms. 7.3.1 Centralized Algorithms and Self-Elective Protocols Many centralized algorithms often use graph theoretic properties for clustering. These algorithms deal with the topology of the entire network as a whole, creating structures
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