On covering all cliques of a chordal graph

For a graph G = (V, E), a vertex set X ⊆ V is called a clique if |X| ⩾ 2 and the graph G[X] induced by X is a complete subgraph maximal under inclusion. We say that a vertex set T is a clique-transversal set if T ∩ X ≠ 0 for all cliques X of G, and define the clique-transversal number τc(G) as the minimum cardinality of a clique-transversal set. Let G be the class of chordal graphs with the property that each edge of G is contained in a clique of order ⩾ 4. Tuza (1990) asked if τc(G) ⩽ |G|/4 for all G ∈ G, where |G| denotes the number of vertices of G. Flotow (1992) had constructed infinitely many examples each showing that the answer to this question is negative. In the present note, by modifying these examples, we show that, for each ε > 0, there exists a graph G ∈ G such that τc(G) ⩾ (27 − ε)|G|. We conjecture τc(G) ⩽ 2|G|/7 for all G ∈ G and present a partial result supporting this conjecture.