Abstract
Consider an undirected and unweighted graph \(G=(V,E)\) where V is the set of vertices and E is the set of edges. Let P be a set of pairs of vertices in G. For a given integer k, we assume that there exists at least k edge disjoint paths between pairs of vertices in P. We consider a new problem that studies the role played by each edge in G to maintain at least k edge disjoint paths between pairs of vertices in P. Hereafter, we refer to this as pairwise k-edge connectivity problem. We compute the Banzhaf power index of each edge to tackle pairwise k-edge connectivity problem that determines the value that edge plays in establishing at least k edge disjoint paths between vertex pairs in P. In particular, we show that computing the Banzhaf power indices of the edges for pairwise k-edge connectivity is \(\#P\)-complete. We also derive a closed form expression for the Banzhaf power index in the k-edge connectivity problem when the graph is a tree.
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Narayanam, L.S.V., Motammanavar, S.V. On computing Banzhaf power index for k-edge connectivity in graphs. Soc. Netw. Anal. Min. 12, 160 (2022). https://doi.org/10.1007/s13278-022-00985-7
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DOI: https://doi.org/10.1007/s13278-022-00985-7