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, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs and proper interval graphs.

翻译：图中的一个团横贯是相交于所有极大团的顶点集。确定最小团横贯大小的问题在文献中受到了广泛关注。本文首次研究了该参数的对偶变体——上团横贯数，定义为最小团横贯的最大可能大小。我们从算法与复杂性角度对该参数进行了探讨，重点考察了多种图类。研究表明，在弦图、弦二部图以及二部图的线图类中，相应的判定问题是NP完全的，但在分裂图与真区间图类中可在线性时间内解决。