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In network design, the all-terminal reliability maximization is of paramount importance. In this classical setting, we assume a simple graph with perfect nodes but independent edge failures with identical probability ρ. The goal is to communicate n terminals using e edges, in such a way that the connectedness probability of the resulting random graph is maximum. A graph with n nodes and e edges that meets the maximum reliability property for all ρ ∈ (0, 1) is called uniformly most-reliable (n, e)-graph (UMRG). The discovery of these graphs is a challenging problem that involves an interplay between extremal graph theory and computational optimization. Recent works confirm the existence of special cubic UMRGs, such as Wagner, Petersen and Yutsis graphs, and a 4-regular graph H = C7. In a foundational work in the field, Boesch. et. al. state with no proof that the bipartite complete graph K4,4 is UMRG. In this paper, we revisit the breakthroughs in the theory of UMRG. A simple methodology to determine UMRGs based on counting trivial cuts is presented. Finally, we test this methodology to mathematically prove that the complete bipartite graph K4,4 is UMRG.

Critical nodes play a major role in network connectivity. Identifying them is important to design efficient strategies to prevent malware or epidemics spread through a network. In this context, the Stochastic Weighted Graph Fragmentation Problem (SWGFP) is a combinatorial optimization problem that belongs to the N P − Complete class. Its objective consists in minimizing the impact of a random attack on a singleton, choosing appropiately a set of nodes to immunize given a restricted budget. In the SWGFP, it is assumed that the attack follows a known probability law and that it affects the whole connected component of the attacked node. In this thesis, a GRASP enriched with Path Relinking algorithm is developed to solve the SWGFP. Its performance is studied under three attack scenarios and compared with a GRASP variant that was previously developed in literature and with a Random heuristic for the problem that picks a set of nodes uniformly at random. Computational experiments show that the algorithm based on Independent Sets which is developed in this thesis, outperforms the other two, with lower expected loss scores and higher robustness.

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