IS γα≤ 5 FOR GRAPHS WHICH HAVE DIAMETER TWO？

A subset of S of the vertex set of a graph G is called acyclic if the subgraph it induces in G contains no cycles. S is called an acyclic dominating set of G if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by γα(G), is called the acyclic domination number of G. S. M. Hedetniemi et al. on 2000 introduced the concept of acyclic domination and posed the following open problem: Is γα(G) ≤ δ(G) for any graph whose diameter is two? In this paper, we give a counterexample which disproves the problem.