Pre-Assignment Problem for Unique Minimum Vertex Cover on Bounded Clique-Width Graphs

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 polynomial time on split graphs and unit interval graphs.