The complexity of reliability computations in planar and acyclic graphs
SIAM Journal on Computing
The Combinatorics of Network Reliability
The Combinatorics of Network Reliability
Chromatic, Flow and Reliability Polynomials: The Complexity of their Coefficients
Combinatorics, Probability and Computing
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
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Consider an undirected multigraph G = (V,E) with n vertices and m edges, and let Ni denote the number of connected spanning subgraphs with i(m ≥ i ≥ n) edges in G. Recently, we showed in [3] the validity of (m-i + 1)Ni-1 (i-n + ⌊3+√9+8(i-n)/2⌋)Ni for a simple graph and each i(m ≥ i ≥ n). Note that, from this inequality, (m-n)Nn/2Nn+1 + Nn/(m-n+1)Nn-1 ≥ 2 is easily derived. In this paper, for a multigraph G and all i(m ≥ i ≥ n), we prove (m-i + 1)Ni-1 ≥ (i-n + 2)Ni, and give a necessary and sufficient condition by which (m-i + 1)Ni-1 = (i-n + 2)Ni. In particular, this means that (m-i + 1)Ni-1 (i-n + ⌊3+√9+8(i-n)/2⌋)Ni is not valid for all multigraphs, in general. Furthermore, we prove (m-n)Nn/2Nn+1 + Nn/(m-n+1)Nn-1 ≥ 2, which is not straightforwardly derived from (m-i + 1)Ni-1 ≥ (i-n+2)Ni, and also introduce a necessary and sufficent condition by which (m-n)Nn/2Nn+1 + Nn(m-n + 1)Nn-1 = 2. Moreover, we show a sufficient condition for a multigraph to have Nn2 Nn-1Nn+1. As special cases of the sufficient condition, we show that if G contains at least ⌈2/3(m-n)⌉+1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then Nn2 Nn-1Nn+1.