Dominating sets in planar graphs

Motivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs. Specifically, we wish to find the smallest ε such that, for n sufficiently large, every n-vertex planar graph contains a dominating set of size at most εn. We prove that 1/4 ≤s ε ≤ 1/3, and we conjecture that ε = 1/4. For triangulated discs we obtain a tight bound of ε = 1/3. The upper bound proof yields a linear-time algorithm for finding an (n/3)-size dominating set.