Finding shortest non-separating and non-disconnecting paths

TL;DR

This paper investigates the computational complexity of finding shortest non-separating and non-disconnecting paths in graphs, revealing hardness results and fixed-parameter tractability for various graph classes and parameters.

Contribution

It provides complexity classifications for shortest non-separating and non-disconnecting path problems, including W[1]-hardness, NP-hardness on specific graph classes, and fixed-parameter tractability results.

Findings

Non-separating path problem is W[1]-hard parameterized by length.

Non-disconnecting path problem is fixed-parameter tractable.

Shortest non-separating path is NP-hard on bipartite, split, and planar graphs.

Abstract

For a connected graph $G = (V, E)$ and $s, t \in V$, a non-separating $s$-$t$ path is a path $P$ between $s$ and $t$ such that the set of vertices of $P$ does not separate $G$, that is, $G - V(P)$ is connected. An $s$-$t$ path is non-disconnecting if $G - E(P)$ is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating $s$-$t$ path of length at most $k$ is W[1]-hard parameterized by $k$, while the non-disconnecting counterpart is fixed-parameter tractable parameterized by $k$. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is fixed-parameter tractable parameterized by $k$ on planar graphs and polynomial-time solvable on chordal graphs if $k$ is the shortest path distance between $s$ and $t$.