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

TL;DR

PAU-VC asks for a minimum pre-assignment S so that G has a unique minimum vertex cover containing S. The authors show PAU-VC is fixed-parameter tractable when parameterized by clique-width, via a bottom-up dynamic program on k-labeled graphs that maintains, for each I ⊆ [k], the size of a minimum vertex cover with respect to I and a collection of minimum β-sets, with a merging procedure for F = (G,lab_G) ×_M (H,lab_H) running in time O(27^{2^k}|V(G)|) and yielding an overall O(3^{2^k}|V(G)|) time given a k-expression. They also derive linear-time algorithms for unit interval graphs and split graphs, and a polynomial-time algorithm for PAU-VC on trees, constituting a significant improvement over prior exponential-time bounds for trees. These results collectively provide efficient tools for generating benchmarks with unique minimum vertex covers and illustrate the power of clique-width-based DP approaches for problems with uniqueness constraints, with implications for AI evaluation datasets and understanding structural graph complexity.

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.

Figures (3)

• Figure 1: An illustration of a clique-width $3$-expression of $P_4$.
• Figure 2: An illustration of the algorithm for a unit interval graph $G$. Figures (a) and (b) represent the unit interval graph $G$ and its representation, respectively. The algorithm returns $\{I_{1,1}, I_{2,2}\}$ as an optimal solution of PAU-VC of $G$, where $\{I_{1,1}, I_{2,2}\}= S[3,3]$ as $S[1,2]=\{I_{1,1}\}, A_{1,2,3}=\{I_{2,2}\},$ and $A_{2,3,3}=\emptyset$. Precisely, $S[3,3]=S[1,2]\cup A_{1,2,3}\cup A_{2,3,3}$.
• Figure 3: $(N_G(b^*)\setminus \{v\})\cup \{b^*\}$ is the minimum vertex set so that exactly one minimum vertex cover $(A\setminus \{v\})\cup \{b^*\}$ in the split graph includes it.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Johansson Johansson1998
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Claim 3.3
• Claim 3.4
• Lemma 3.5