Claw-free cubic graphs with clique-transversal number half of their order

A clique-transversal set  DD of a graph GG is a set of vertices of GG such that DD meets all cliques of GG. The clique-transversal number  , denoted by τC(G)τC(G), is the minimum cardinality of a clique-transversal set in GG. In 2008, we showed that the clique-transversal number of every claw-free cubic graph is bounded above by half of its order. In this note we characterize claw-free cubic graphs which attain the upper bound.