Computing the K-terminal Reliability of Circle Graphs

TL;DR

This paper investigates the computational complexity of K-terminal reliability in circle graphs, proving it remains #P-complete, and introduces a linear-time algorithm for proper circular-arc graphs, a subclass of circle graphs.

Contribution

It establishes the #P-completeness of K-terminal reliability for circle graphs and presents a linear-time algorithm for proper circular-arc graphs, expanding the understanding of efficient computation in these classes.

Findings

K-terminal reliability is #P-complete for circle graphs.

A linear-time algorithm exists for proper circular-arc graphs.

The complexity status for circle graphs is confirmed as #P-complete.

Abstract

Let G denote a graph and let K be a subset of vertices that are a set of target vertices of G. The K-terminal reliability of G is defined as the probability that all target vertices in K are connected, considering the possible failures of non-target vertices of G. The problem of computing K-terminal reliability is known to be #P-complete for polygon-circle graphs, and can be solved in polynomial-time for t-polygon graphs, which are a subclass of polygon-circle graphs. The class of circle graphs is a subclass of polygon-circle graphs and a superclass of t-polygon graphs. Therefore, the problem of computing K-terminal reliability for circle graphs is of particular interest. This paper proves that the problem remains #P-complete even for circle graphs. Additionally, this paper proposes a linear-time algorithm for solving the problem for proper circular-arc graphs, which are a subclass of circle graphs and a superclass of proper interval graphs.