The Upper Clique Transversal Problem
Martin Milanič, Yushi Uno
TL;DR
The paper defines the upper clique transversal number $\tau_c^+(G)$ as the maximum size of a minimal clique transversal and studies the UCT decision problem. It links UCT to induced matchings via the vertex-clique incidence graph $B_G$, enabling linear-time algorithms for split graphs, proper interval graphs, and cographs, and a MSO$_1$-based approach for bounded cliquewidth. It proves NP-completeness for chordal, chordal bipartite, cubic planar bipartite, and line graphs of bipartite graphs, while providing polynomial-time results for graphs of bounded cliquewidth, thereby mapping the complexity landscape. The work employs reductions from Spanning Star Forest and Independent Dominating Set, cotree-based DP for cographs, and induced-matching techniques to derive the tractability boundaries, and it raises open questions about interval graphs and other width parameters. The results advance understanding of upper variants of classical clique problems and open avenues for width-parameter and parameterized analyses.
Abstract
A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the ''upper'' variant of this parameter, the upper clique transversal number, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, cubic planar bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs, proper interval graphs, and cographs, and in polynomial time for graphs of bounded cliquewidth. We conclude the paper with a number of open questions.