Shortest non-separating st-path on chordal graphs

TL;DR

This paper demonstrates that finding the shortest non-separating st-path is NP-Hard on general graphs but can be efficiently solved in linear time on chordal graphs, providing a practical algorithm for such cases.

Contribution

It introduces a linear-time algorithm for finding the shortest non-separating st-path on chordal graphs, contrasting with NP-Hardness on general graphs.

Findings

NP-Hardness of the problem on general graphs

Linear-time algorithm for chordal graphs

Efficient shortest path computation on chordal graphs

Abstract

Many NP-Hard problems on general graphs, such as maximum independence set, maximal cliques and graph coloring can be solved efficiently on chordal graphs. In this paper, we explore the problem of non-separating st-paths defined on edges: for a connected undirected graph and two vertices, a non-separating path is a path between the two vertices such that if we remove all the edges on the path, the graph remains connected. We show that on general graphs, checking the existence of non-separating st-paths is NP-Hard, but the same problem can be solved in linear time on chordal graphs. In the case that such path exists, we introduce an algorithm that finds the shortest non-separating st-path on a connected chordal graph of $n$ vertices and $m$ edges with positive edge lengths that runs in $O(n\log{n} + m)$ time.