Greedy domination on biclique-free graphs

The greedy algorithm for approximating dominating sets is a simple method that is known to compute an $(\ln n+1)$-approximation of a minimum dominating set on any graph with $n$ vertices. We show that a small modification of the greedy algorithm can be used to compute an $O(t^2\cdot \ln k)$-approximation, where~$k$ is the size of a minimum dominating set, on graphs that exclude the complete bipartite graph $K_{t,t}$ as a subgraph.