  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.
