# Network reliability as a result of redundant connectivity

There exists, for any connected graph G, a minimum set of vertices that, when removed, disconnects G. Such a set of vertices is known as a minimum cut-set, the cardinality of which is known as the connectivity number k(G) of G. A connectivity preserving [connectivity reducing, respectively] spanning subgraph G0 ? G may be constructed by removing certain edges of G in such a way that k(G0) = k(G) [k(G0) < k(G), respectively]. The problem of constructing such a connectivity preserving or reducing spanning subgraph of minimum weight is known to be NP–complete. This thesis contains a summary of the most recent results (as in 2006) from a comprehensive survey of literature on topics related to the connectivity of graphs. Secondly, the computational problems of constructing a minimum weight connectivity preserving or connectivity reducing spanning subgraph for a given graph G are considered in this thesis. In particular, three algorithms are developed for constructing such spanning subgraphs. The theoretical basis for each algorithm is established and discussed in detail. The practicality of the algorithms are compared in terms of their worst-case running times as well as their solution qualities. The fastest of these three algorithms has a worst-case running time that compares favourably with the fastest algorithm in the literature. Finally, a computerised decision support system, called Connectivity Algorithms, is developed which is capable of implementing the three algorithms described above for a user-specified input graph.

Thesis (MSc (Logistics)--University of Stellenbosch, 2007.

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