A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. It is NP-hard to determine the minimum cardinality τc of a clique-transversal of G. In this work, first we propose an algorithm for determining this parameter for a general graph, which runs in polynomial time, for fixed τc. This algorithm is employed for finding the minimum cardinality clique-transversal of 3K2-free circular-arc graphs in O(n4) time. Further we describe an algorithm for determining τc of a Helly circular-arc graph in O(n) time. This represents an improvement over an existing algorithm by Guruswami and Pandu Rangan which requires O(n2) time. Finally, the last proposed algorithm is modified, so as to solve the weighted version of the corresponding problem, in O(n2) time.
