<inline-formula> <tex-math notation="LaTeX">$K$</tex-math> </inline-formula>-Terminal Reliability of <inline-formula> <tex-math notation="LaTeX">$d$</tex-math> </inline-formula>-Trapezoid Graphs

Abstract

Given a probabilistic graph, with reliable edges and unreliable vertices, $K$ -terminal reliability problem is to find the probability that a given subset $K$ of vertices remains connected. The problem is #P-complete for general graphs, even so for some special graphs such as chordal graphs, and comparability graphs. However, polynomial time algorithms to solve the problem have been designed for some special graphs such as interval graphs, permutation graphs, and $d$ -trapezoid graphs. Existing time complexity of the polynomial time algorithm for the $d$ -trapezoid graph is $O(n^{2d+1})$ , which is a very high degree polynomial. This makes it impractical to solve large problem instances using the existing algorithm. Here, we propose a novel technique and use it to design a simple linear-time algorithm to solve the problem on $d$ -trapezoid graphs. As the complexity of our algorithm is linear, it is no more a difficulty to solve large problem instances in small amount of time.