*3.3. Connectivity Proof*

We show the connectivity of the resulting topology of Algorithm 1 derived from the given deployment of UAVs. We preliminarily assume that any of the deployed UAVs has one or more nearby UAVs in its maximum transmission range. We first prove local connectivity among the neighbors in the maximum transmission range of UAV. Then, we eventually prove global connectivity by finding a knock-on path from two arbitrary UAVs *u* to *v*, referred to *P*(*u*, *v*) =< *u*, *h*1, *h*2, *h*3, ... , *hm*, *v* >. This global connectivity is significantly derived from UAVs' previous local connections. In terms of the resulting topology of our system, we derive the following two theorems about connectivity.

**Theorem 1** (Local connectivity)**.** *The result of Algorithm 1 guarantees the connectivity between a UAV and its neighbor UAVs within the maximum transmission range.*

**Figure 6.** Space partition with respect to *n*. (**a**) Space partition when *n* = 4; (**b**) Space partition when *n* = 6; (**c**) Space partition when *n* = 8; (**d**) Space partition when *n* = 12; (**e**) Space partition when *n* = 20.

**Proof of Theorem 1.** This theorem shows the UAV connectivity between an arbitrary UAV *u* and its neighbor UAVs, all of which are within one maximum transmission range from the UAV *u* itself. Let *MTRu* refer to a set of the UAVs located within the maximum transmission range, where *w* ∈ *MTRu*. Also, let *NHu* refer to a set of selected links by *u* in Algorithm 1, where *NHu* ⊂ *MTRu*.

∀*w* ∈ *MTRu*, **Case** (**i**) : *w* ∈ *NHu P*(*u*, *w*) = < *u*, *w* > **Case** (**ii**) : *w* ∈/ *NHu* there always exists *w*<sup>1</sup> such that *w*<sup>1</sup> ∈ *NHu* and *P* =< *u*, *w*<sup>1</sup> > is a subpath of *Pu*(*u*, *w*), which is derived by an MST constructed in line 26 of Algorithm 1. We let *u* = *w*0, and applying Algorithm 1 to each *wi*, then there exists *wk*−<sup>1</sup> such that *wk* = *<sup>w</sup>*. By chaining all the discovered subpaths, *PTotal* = *P* (*w*0(= *<sup>u</sup>*), *<sup>w</sup>*1) <sup>∪</sup> *<sup>P</sup>*(*w*1, *<sup>w</sup>*2) <sup>∪</sup> ... <sup>∪</sup> *<sup>P</sup>k*(*wk*−1, *wk*(= *<sup>w</sup>*)) = *<sup>P</sup>*(*u*, *<sup>w</sup>*).

$$\text{Thus, the path } P(u, w) =  \text{ exists.}$$

Figure 7 graphically represents the above sequence. By this procedure, the connectivity with all UAVs in every partition of arbitrary chosen UAV *u* is always guaranteed. Thus, the connection is guaranteed for all UAVs in the maximum transmission range of UAV *u*.

**Figure 7.** Finding routing path of the topology control layer.

**Theorem 2** (Global connectivity)**.** *The result of Algorithm 1 guarantees the connectivity of two arbitrary UAVs network-wide.*

**Proof of Theorem 2.** We prove the global connectivity with Theorem 1. We let *G*(*V*, *E*) refer to the graph representing a fully connected UAV network, where *V* is the set of UAVs and E is the set of edges defined by the reachability between two UAVs with the maximum transmission range. Also, *GTC*(*V*, *ETC*) refers to the graph representing the resulting network topology of topology control layer, where *ETC* is the set of edges that Algorithm 1 selects. The other notations mean the same as aforementioned.

Assume there is a path *P*(*u*, *v*) = < *u*(= *h*0), *h*1, *h*2,..., *v*(= *hn*) > in *G*(*V*, *E*). Each pair of subsequent UAVs in *P*(*u*, *v*), such as (*hi*, *hi*+1)(*i* = 0, 1, . . . , *n* − 1), is replaced with a path < *hi*, *<sup>w</sup>*1, *<sup>w</sup>*2,..., *wk*−1, *hi*+<sup>1</sup> >, owing to Theorem 1. Therefore, if *P*(*u*, *v*)is in *G*(*V*, *E*), then *P*(*u*, *v*)is also in *GTC*(*V*, *ETC*).
