Abstract
We study various ideals arising in the theory of system reliability. We use ideas from the theory of divisors, orientations, and matroids on graphs to describe the minimal polyhedral cellular resolutions of these ideals. In each case, we give an explicit combinatorial description of the minimal generating set for each higher syzygy module in terms of the acyclic orientations of the graph, the reduced divisors, and the bounded regions of the graphic hyperplane arrangement. The resolutions of all these ideals are closely related, and their Betti numbers are independent of the characteristic of the base field. We apply these results to compute the reliability of their associated systems.
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Notes
The Tutte polynomial is defined by \(T(x,y)=\sum _{A\subseteq E}(x-1)^{k(A)-k(E(G))}(y-1)^{k(A)+|A|-|V(G)|}\), where k(A) is the number of connected components of the subgraph on A.
We use the notation \({\mathcal C}_G^s\) in order to be coherent with the notation used in §4; however, in [22], it is denoted by \({\mathcal O}_{{\mathcal {M}}}\), where \({\mathcal {M}}\) is the associated graphic matroid. We let \({\mathcal C}_G^s\), \({\mathcal C}_G\), \({\mathfrak T}_G^s\), and \({\mathfrak T}_G\), respectively, denote the ideals \({\mathcal O}_{{\mathcal H}_G^{s}}\), \(\bar{{\mathcal O}}_{{\mathcal H}_G^{s}}\), \({\mathcal O}_{{\mathcal H}_G^{s}}^\vee \), and \(\bar{{\mathcal O}}_{{\mathcal H}_G^{s}}^\vee \) from §3.1.
Gioan in [14] defines a s-connected orientation as an orientation in which every vertex is reachable from s by a directed path. We use the notation \({\mathfrak T}_k\) and the name oriented k-spanning tree in order to emphasize that \({\mathfrak T}_k\) is an acyclic subgraph of G containing a rooted spanning tree with k extra edges.
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Acknowledgments
The author is very grateful to Bernd Sturmfels and Volkmar Welker for many helpful conversations, and she would like to thank Lionel Levine, Dinh Le Van, and Raman Sanyal for their comments on the first draft. She also thanks Eduardo Sáenz-de-Cabezón and Henry Wynn for introducing her to system reliability theory. The author was supported by the Alexander von Humboldt Foundation.
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Mohammadi, F. Divisors on graphs, orientations, syzygies, and system reliability. J Algebr Comb 43, 465–483 (2016). https://doi.org/10.1007/s10801-015-0641-y
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DOI: https://doi.org/10.1007/s10801-015-0641-y