Cut paths and their remainder structure, with applications

TL;DR

This paper introduces the concept of cut paths as a generalization of cut arcs in strongly connected graphs, providing efficient algorithms for their verification and applications to reachability problems in bioinformatics.

Contribution

It defines cut paths and their remainder structures, offers linear-time algorithms for verifying safety and multi-safety, and improves the computation of multi-safe walks.

Findings

Linear-time verification of cut paths.

First linear-time algorithm for multi-safety.

Improved algorithm for computing all maximal multi-safe walks.

Abstract

In a strongly connected graph $G = (V,E)$, a cut arc (also called strong bridge) is an arc $e \in E$ whose removal makes the graph no longer strongly connected. Equivalently, there exist $u,v \in V$, such that all $u$-$v$ walks contain $e$. Cut arcs are a fundamental graph-theoretic notion, with countless applications, especially in reachability problems. In this paper we initiate the study of cut paths, as a generalisation of cut arcs, which we naturally define as those paths $P$ for which there exist $u,v \in V$, such that all $u$-$v$ walks contain $P$ as subwalk. We first prove various properties of cut paths and define their remainder structures, which we use to present a simple $O(m)$-time verification algorithm for a cut path ($|V| = n$, $|E| = m$). Secondly, we apply cut paths and their remainder structures to improve several reachability problems from bioinformatics. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. We show that cut paths provide simple $O(m)$-time algorithms verifying if a walk is safe or multi-safe. For multi-safety, we present the first linear time algorithm, while for safety, we present a simple algorithm where the state-of-the-art employed complex data structures. Finally we show that the simultaneous computation of remainder structures of all subwalks of a cut path can be performed in linear time. These properties yield an $O(mn)$ algorithm outputting all maximal multi-safe walks, improving over the state-of-the-art algorithm running in time $O(m^2+n^3)$. The results of this paper only scratch the surface in the study of cut paths, and we believe a rich structure of a graph can be revealed, considering the perspective of a path, instead of just an arc.