A linear kernel for a planar connected dominating set

We provide polynomial time data reduction rules for Connected Dominating Set on planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm analyzes the input graph and either finds an appropriate reduction rule that can be applied, or zooms in on a region of the graph which is more amenable to reduction. We find this method of independent interest and believe that it will be useful for obtaining linear kernels for other problems on planar graphs.