Triconnected decomposition for computing K-terminal network reliability

Let R(Gk) denote the probability that a subset of vertices K in the undirected graph G = (V, E) can communicate when the edge of G fail independently with known probabilities. Suppose that Gk contains as separting pair of vertices {u, v} such that Gk= Gk ∪ Ǧǩ, Ṽ ∩ V ={u v}, Ẽ ∩ E = o, |Ẽ| ⩾ 2, and |E| ⩾2. It is shown that Ĝk may be replaced by a chain χ of at most three edges between u and v such that R(Gk) = ΩR ((G ∪ χ)k′) where Ω is a derived constant and K′ is defined by the procedure. The failure probabilities of the edges in χ and the value of Ω are shown to be computable by evaluating the reliability of one of four variants of Ĝk. Minimal components Ĝk can be efficiently found using triconnected decomposition and the above procedure embedded within a recursive factoring algorithm for computing K-terminal reliability. Conditions are derived which guarantee the new algorithm is at least as good as a pure factoring algorithm, and an example shows that the new algorithm can be exponentially better.