Pre-assignment problem for unique minimum vertex cover on bounded clique-width graphs

TL;DR

This paper studies the problem of finding a minimal vertex set that guarantees a unique minimum vertex cover in graphs, demonstrating fixed-parameter tractability with respect to clique-width and efficient solutions for specific graph classes.

Contribution

It introduces the PAU-VC problem, proves its fixed-parameter tractability for graphs with bounded clique-width, and provides linear-time algorithms for split and unit interval graphs.

Findings

PAU-VC is fixed-parameter tractable parameterized by clique-width.

Linear-time algorithms exist for PAU-VC on split graphs.

Linear-time algorithms exist for PAU-VC on unit interval graphs.

Abstract

Horiyama et al. (AAAI 2024) considered the problem of generating instances with a unique minimum vertex cover under certain conditions. The Pre-assignment for Uniquification of Minimum Vertex Cover problem (shortly PAU-VC) is the problem, for given a graph $G$, to find a minimum set $S$ of vertices in $G$ such that there is a unique minimum vertex cover of $G$ containing $S$. We show that PAU-VC is fixed-parameter tractable parameterized by clique-width, which improves an exponential algorithm for trees given by Horiyama et al. Among natural graph classes with unbounded clique-width, we show that the problem can be solved in linear time on split graphs and unit interval graphs.