## 4 Conclusions and Future Research Directions  
In this paper, we propose a method for designing resilient backhaul topologies, based on graph invariants. We also proposed a new invariant, the node Wiener impact, which measures the impact of a node failure to the distances among the remaining node pairs. Besides, we are currently working on a version of the node Wiener impact invariant for analyzing the impact of link failures. As demonstrated herein, our graph theoretic approach to backhaul topology design enables the systematic refinement of backhaul topologies on the basis of wireless network objectives. Furthermore, our method enables the direct comparison of topologies and examination of the trade-offs in network design. Moreover, this manner of designing network topologies allows us to bring the large knowledge base available within the graph theory community to overcome telecommunications challenges. In summary, our solution could provide guidance in the decision making about the impacts on the network when new entities (edges and nodes) are added or removed in the topology. For the C-RAN, this is fundamental to raising the survivability and making feasible the usage of the backhaul as part of the C-RAN underlying infrastructure. This work can be extended in many ways. For instance, to implement the interconnection solutions obtained by our method, an important practical aspect to be further studied is the embedding of the nodes of a improved solution into the nodes of the original solution. In addition, one could use the interconnection solution as the basis to design an ILP model that takes into account traffic demands. Another interesting investigation direction is the extension of our method for multi-layered network design. In C-RAN context, we devise problems with the sharing of physical resources, such as fibers and optical equipment, between backhaul and fronthaul networks. In this scenario, two overlay networks with distinct objectives impact the physical topology planning. An additional future objective is to explore how our method behaves in dynamically allocated scenarios, such as heterogeneous cloud radio access network (H-CRAN) [7]. In this problem, we must consider the on/off switching of the RRH unities, and the dynamic BBUs and RRHs re-association. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Invariants** & **Original** & **Addition of Two Edges** & **Minimization of Max. WI** \\ \hline Nr. of edges & 33 & 33 & 34 \\ Max. degree & 6 & 6 & 5 \\ Average degree & 2.75 & 2.75 & 2.83 \\ Degree variance & 1.85 & 1.52 & 0.89 \\ Diameter & 9 & 6 & 6 \\ Average distance & 3.55 & 3.06 & 2.91 \\ Wiener index & 979 & 844 & 802 \\ Max./Min. edge betweenness & 97.5/6.58 & 53.27/7.00 & 39.38/10.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: RNP backbone after the worst case node failure (max. WI). Figure 7: Invariants after the worst case node failure (max. WI): (**a**) diameter; and (**b**) average distance. In this direction, our proposal can also benefit from the SDN/NFV paradigm for managing and orchestrating network topology according to dynamic demands [40]. Following the philosophy of funded collaborative research project, this work was a joint contribution of all mentioned authors. The efforts to devise, implement and produce the results were spent during the work of the EU-Brazil co-funded research project called FUTEBOL (Federated Union of Telecommunications Research Facilities for an EU-Brazil Open Laboratory) [41]. In general, the following can be said: writing--original draft, C.K.D., M.H.M.P., G.C., M.A.M., M.R.N.R., M.E.V.S., M.M., M.E.M. and C.B.B. ; and writing--review and editing, V.F. This work was part of the FUTBOL project, which has received funding from the European Union's Horizon 2020 for research, technological development, and demonstration under grant agreement No. 688941 (FUTEBOL), as well from the Brazilian Ministry of Science, Technology and Innovation (MCTI) through RNP and CTIC.