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

Abstract

Horiyama et al. (AAAI 2024) considered the problem of generating instances with a unique minimum vertex cover under certain conditions. The Minimum Pre-assignment for Uniquification of Minimum Vertex Cover problem (shortly Min PAU-VC) is the problem, for given a graph $G$, to find a minimum set $S$ of vertices in $G$ such that among all minimum vertex covers of $G$, exactly one contains $S$. We show that Min 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.

20 pages, 3 figures