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 smallestεsuch that, fornsufficiently large, everyn-vertex planar graph contains a dominating set of size at mostεn.We prove that 1/4<ε<1/3, and we conjecture thatε=1/4.For triangulated discs we obtain a tight bound ofε=1/3.The upper bound proof yields a linear-time algorithm for finding an(n/3)-size dominating set.