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This superset contains O(mn) cycles which are constructed in the following way. For each vertex v and edge e = (u, w), construct the cycle C = SP (v, u) + SP (v, w) + (u, w) where SP (a, b) is the shortest path from a to b. If these two shortest paths do not contain a vertex other than v in common then keep the cycle otherwise discard it. Let us call this set of cycles the Horton set. It was shown in  that the Horton set always contains an MCB. However, not every MCB is contained in the Horton set.
IEEE Computer Society, 2000. de Abstract. In this paper we consider the problem of computing a minimum cycle basis of an undirected graph G = (V, E) with n vertices and m edges. We describe an eﬃcient implementation of an O(m3 + mn2 log n) algorithm presented in . For sparse graphs this is the currently best known algorithm. This algorithm’s running time can be partitioned into two parts with time O(m3 ) and O(m2 n + mn2 log n) respectively. Our experimental ﬁndings imply that the true bottleneck of a sophisticated implementation is the O(m2 n + mn2 log n) part.
For dense graphs however, we observe a substantial diﬀerence in performance. Dense Unweighted Graphs. In the case of dense unweighted graphs, the hybrid algorithm performs better than the other algorithms. However, even on the exact same graph, the addition of weights changes the performance substantially. This change in performance is not due to the diﬀerence in size of the produced Horton set, between the unweighted and the weighted case, but due to the total number of queries that have to be performed in this set.
Airbus A320 SOP 07Descent