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CHECK These messages are periodically sent by the noncoordinator nodes. They are used to remove the potential network partitions. Each CHECK message also contains of list of coordinator neighbors of the node that made it the noncoordinator. CHECK REPLY Upon receipt of a CHECK message, noncoordinator node compares the coordinator neighbor list included in the CHECK message with the neighbor list of the node that made it a noncoordinator. If there are no common entries in the two lists, it sends a CHECK REPLY message.

This can be shown as follows. According to rule 4, if a gray node has white vertices as neighbors, then it is colored black and all its neighbors except the neighboring vertex belonging to the longest edge are colored gray. A black node has all its neighbors colored black or gray according to rule 2. This completes the proof of the theorem. 3 The spanning subgraph G generated by MAXCOVSPAN provides the highest coverage among all spanning subgraphs of G that have the same number of nodes as G . PROOF In order to avoid case-by-case analysis, we prove this theorem using mathematical induction.

For example, the Toff value that triggers the state transition from an undecided state to an ETNC state depends on the radio range of the specific sensor used in the sensor network. Time Relationships are as follows: The relationships between Toff , Truntime , Tsetup , Tcoord , and Tnoncoord 1. Toff < Tcoord < Tsetup . 2. Tcoord < Tsetup and Tnon−coord < Tsetup . 4 Tr T r = T runtime Illustration of the relationships between the time intervals. 4. 5 shows the result of applying SCARE to an example sensor network with 100 randomly deployed nodes in a 100-m × 100-m grid.