Domination Reliability
Klaus Dohmen, Peter Tittmann
TL;DR
The paper introduces domination reliability as a novel graph-based reliability measure driven by dominating sets under independent vertex failures, linking it to the domination polynomial and hypergraph coverage. It develops decomposition, inclusion-exclusion, and neighborhood-graph techniques to compute $DRel$, and defines the domination reliability polynomial for equal vertex probability $p$ along with explicit results for several graph families. A key contribution is the demonstrated connection between domination reliability and the domination polynomial, including analogues of broken circuit theorems and multiple equivalent formulations. The authors prove NP-hardness of computing domination reliability and establish a bidirectional equivalence with hypergraph coverage probability, enabling a cross-pollination between graph polynomials and system reliability. Overall, the work lays a formal foundation for studying domination-based reliability and its polynomial invariants, with practical implications for modeling coherent binary systems via hypergraphs.
Abstract
We propose a new network reliability measure for some particular kind of service networks, which we refer to as domination reliability. We relate this new reliability measure to the domination polynomial of a graph and the coverage probability of a hypergraph. We derive explicit and recursive formulae for domination reliability and its associated domination reliability polynomial, deduce an analogue of Whitney's broken circuit theorem, and prove that computing domination reliability is NP-hard.