Finding the most vital edge with respect to minimum spanning tree in weighted graphs

Parallel Algorithm for Finding the Most Vital Edge in Weighted Graphs

“…If e is an edge of MST(G), and G − e is connected the replacement edge r(e) [11] is defined to be the edge such that MST(G) − e + r(e) is a minimum spanning tree of G − e. If G − e is not connected the r(e) is undefined. As the most vital edge of G belongs to MST(G) [8], for a bridgeless graph G the most vital edge is that edge which maximises w(r(e))−w(e) among e ∈ MST(G).…”

“…A most vital edge is an edge, which if removed, causes the maximum change (increase) in the cost of the minimum spanning tree (MST(G)) of the graph G. The problem of finding the most vital edge can be solved sequentially in O(m log n) and O(n 2 ) time [8]; in fact the time can be improved to O(m + n log n) and O(mα(m, n)) [11]. The problem can be solved in O(log n) time using n 2 / log n processors on an exclusive read exclusive write (EREW) parallel random access machine (PRAM) [9].…”

Increasing the Weight of Minimum Spanning Trees

“…The discrete version of our problem has appeared in the literature under the name of the most vital edges in the minimum spanning tree problem [15,16,18,211. In 1211 it is shown that when the edges of a graph have arbitrary destruction costs, the problem of computing the maximum increase in the weight of the minimum spanning trees achievable by removing edges of a certain total destruction cost is NP-hard.…”

“…In 1211 it is shown that when the edges of a graph have arbitrary destruction costs, the problem of computing the maximum increase in the weight of the minimum spanning trees achievable by removing edges of a certain total destruction cost is NP-hard. In [15,16,18, 281 algorithms are given for finding the single most vital edge for a minimum spanning tree. The continuous version of the robustness problem for minimum spanning trees can be seen as a generalization of the sensitivity analysis for minimum spanning trees 1281, since we need to consider simultaneous changes in the weights of several edges.…”

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

“…Algorithms for the most vital edge problems on shortest paths include Bar-Noy et al (1995), Malik et al (1989), Venema et al (1996). Algorithms for finding the most vital edges in minimum cost spanning trees have been developed by Hsu et al (1991), Hsu et al (1992), Iwano and Kato (1993), and Banerjee and Saxena (1997). The most vital edge problem for minimum cost spanning trees is not directly related to the problem of identifying edge tolerances, although both are fundamental questions of sensitivity analysis.…”