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 ffl such that, for n sufficiently large, every n-vertex planar graph contains a dominating set of size at most ffln. We prove that 1 4 ffl 1 3 , and we conjecture that ffl = 1 4 . For triangulated discs we obtain a tight bound of ffl = 1 3 . The upper bound proof yields a linear-time algorithm for finding an n/3 - size dominating set