put a sensor in one of that points and repeat this until all the region A **be at least 1-covered, we need to put** 
at most 6 new sensors in the region A **, because we** 
put a sensor in a point that not covered by any other sensor and so it's distance from all other nodes is at least Rs
, so the distance between any pair of new sensors is at least Rs
, also all of this sensors be in a circle with radius Rs
,because before eliminating node u , all the region A **was at least 1-covered, so** 
the points that become uncovered were covered by node u **, also none of this new sensor be in the center** 
of circle that cover all of them (C(u) **) because the** 
distance between node u and node v **is less than** 
Rs
, so the position of node u **is covered by node** v , 
now according to lemma 3; we need to put at most 6 new sensors. With this method there is no new pair of sensors that the distance between them is less than Rs
, because the positions of new sensors be uncovered before putting these new sensors and so the distance between the position of these new sensors and any other sensor is at least Rs
. 

By this method we eliminate one pair of sensors that distance between them is less than Rs and because the number of sensors and so the number of these pairs is finite, finally we reach a distribution that the distance between any pair of sensors is at least Rs and the region A **is at least 1-covered.** 

## 5. Conclusions And Future Works

In this paper we present a tighter sufficient condition for 1-coverage to imply connectivity in wireless sensor networks and corresponding sensor location method, which can facilitate the sensor location problem and save the energy. We put a constraint on the minimum allowed distance between any two sensors, this constraint is not a restrictive condition because we present a new distribution method for sensors with sensing range Rs
; which previously cover a convex region A **; that the distance between any two sensor is at** 
least Rs
,in this method we put at most 6 new sensors instead of any previous sensor, but in most practical application there is no need to put 6 new sensors, and few sensors are enough. also we consider a rectangular region A = a *b **, such that** 
Rs ≤ a,Rs ≤ b **, all the proofs presented work for a** 
convex polygonal region that any angle of it is at least !

60 **, and length of all sides of it are at least** 
Rs
, almost all the real regions that should be covered satisfy these conditions and so this is not a restrictive constraint. By this result we can use sensors with shorter communication range, so these sensors because need to send messages to shorter distance consume less energy. 

In the future, we will extend our solution to handle other convex regions and other constraints on minimum allowed distance between sensors, and make our condition tighter and save much energy. 

## 6. References

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of ACM MobiCom'99, 1999. 

[3] A Mainwaring, J Polastre, R Szewczyk and D Culler, 
"Wireless sensor networks for habitat monitoring"**, In First** 
ACM International Workshop on Wireless Workshop in Wireless Sensor Networks and Applications (WSNA 2002), 
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April 2001. 

[5] S. Meguerdichian, F. Koushanfar, G. Qu, and M. Potkonjak, **"Exposure in Wireless Ad Hoc Sensor** 
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on Mobile Computing and Networking (MobiCom'01), pp. 

139-150, July 2001. 

[6] S. Meguerdichian and M. Potkonjak. *"Low Power 0/1* Coverage and Scheduling Techniques in Sensor Networks."**, UCLA Technical Reports 030001. January** 
2003.

[7] X Wang, G Xing, Y Zhang, C Lu, R Pless and C Gill 
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[8] Honghai Zhang and Jennifer C. **" Maintaining Sensing** 
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