higher sensing accuracy and be more robust against sensing failures. The coverage configuration problem can be formulated as follows. Given a convex coverage region A **, and a coverage degree** K
specified by the application (either before or after deployment), we must maximize the number of sleeping nodes under the constraint that the remaining nodes must guarantee A **is K-covered.** 
In addition, we assume that any two nodes u **and** v can directly communicate with each other if their Euclidian distance is less than a communication range Rc
, *i.e.*, uv < Rc
. Given a coverage region A and a sensor coverage degree K **, the goal of an** 
integrated coverage and connectivity configuration is maximizing the number of nodes that are scheduled to sleep under the constraints that the remaining nodes must guarantee: 1) A is at least K **–covered,** 
and 2) all active nodes are connected. 

## 3. Sufficient Condition For 1-Coverage To Imply Connectivity

Define the graph G(V,E) **to be the communication** 
graph of a set of sensors, where each sensor in the set is represented by a node inV **, and for any node** x and y in V , the edge (x, y) exist in E **if and only** if the Euclidean distance between x and y **is less** 
than Rc
, *i.e.*, Rc xy < . Node v and u **are** 
connected in G(V,E) if and only if a *network path* consisting of consecutive edges in E **exists between** 
node u and v **, we put a constraint on minimum** 
allowed distance between any two nodes, the minimum allowed distance between any two nodes is Rs
. 

Now we assume a scenario like this (Figure 1.): We assume a rectangular region A = a *b **, such** 
that Rs ≤ a,Rs ≤ b **, we assume this region has a** 
complete coverage, for any two nodes u **and** v **in** 
region A , let Puv **be the line segment joining them.** Since region A is convex, Puv **remains entirely** within A . Hence any point on Puv **is at least 1-**
covered. consider the intersection point of C(u) **and** 
Puv and name this point as w**, according to our** 
definition, this point not covered by u **, We name the** 
node that covers w as x **, also we name the** 
intersection points of C(u) **and** C(x) **as** s,t . 

![0_image_0.png](0_image_0.png)

Proof. x covers w **so the Euclidian distance** 
between x and w **is less than** Rs
. 

xw < Rs = uw **. So according to triangle** 
inequality we have: 
xv < xw + wv < uw + wv = uv , If x **lie on** 
the Puv **The above inequality is apparent.** 
Lemma 2. **If** 2 + 3 *Rs ≤ Rc then node u *is* connected to node x in G(V,E) . 

Proof. **(by contradiction) If the Euclidian distance** 
between x and u **is less than** Rc then the lemma is correct, so we assume that R ux c ≤ **. In this case the** 
distance between point s and t **should be less than** 
Rs because otherwise if this distance be at least Rs
, 
the angles ∠u in triangle tus and ∠x **in triangle** 
 txs **are at least** !

60 (sides tu and su **in triangle** 
 tus, and sides tx and sx in triangle txs **, are** 
all equal to Rs
; so according to Law of sines in triangles(C
c B
b A
a sin sin sin
= = **), angles** ∠u in triangle tus and ∠x in triangle txs **must be at** 
least !

60 ) so the angle ∠t in triangle utx **is at** 
most !

120 **(because of symmetry the line** ux **is** 
bisector of angles ∠u in triangle tus **and** ∠x in triangle txs so the angles ∠u and ∠x **in triangle** 
 utx **are at least** !

30 **and so because the sum of** 
three angles of every triangle is !

180 **the angle** 