## 1 Introduction
Given a graph \(G\) and a branch-decomposition of \(G\) with width \(\beta\), FT algorithm finds an optimal solution in \(O(2^{(3\log_{4}3)\beta}n)\) time for the dominating set problem. For a planar graph \(G\), it is known that a branch-decomposition of minimum width \(\mathrm{bw}(G)\) can be computed in \(O(n^{3})\) time [17,18] and \(\mathrm{bw}(G)\leq 3\sqrt{4.5\gamma(G)}\)[14,19]. [PERSON] _et al_. [12] give an \(O(n^{3})\) time algorithm which computes a subgraph \(H\) (kernel) of \(G\) such that \(H\) has \(O(\gamma(G))\) vertices, \(\gamma(H)\leq\gamma(G)\)and a minimum dominating set of \(G\) can be constructed from a minimum dominating set of \(H\) in linear time. Notice that for a subgraph \(H\) of \(G\), \(\mathrm{bw}(H)\leq\mathrm{bw}(G)\). From the above, the FT algorithm solves the planar dominating set problem in \(O(2^{(3\log_{4}3)\mathrm{bw}(G)}\gamma(G)+n^{3})\) and \(O(2^{15.13\sqrt{\gamma(G)}}\gamma(G)+n^{3})\) time (The running time of FT algorithm can be further improved to \(O(2^{11.98\sqrt{\gamma(G)}}\gamma(G)+n^{3})\) using fast matrix multiplication in the dynamic programming step of the algorithm [20]. However, this improvement is only of theoretical interest because the fast matrix multiplication is not practical [21]). For graphs with small treewidth/branchwidth, an FPT algorithm may be efficient to find an optimal solution, however, for graphs with large treewidth/branchwidth, one may have to rely on approximation algorithms for the planar dominating set problem. A PTAS is highly desired if the solution values are required to be close to optimal with a guaranteed approximation ratio. [PERSON] introduces a framework to obtain PTAS for a class of NP-hard problems [9]. This framework is based on decomposing a planar graph into \(k\)-outer planar subgraphs. A graph \(G\) is called _outer planar_ or _1-outer planar_ if it has a planar embedding such that all vertices of \(G\) are incident to a same face (called outer face). For \(k>1\), \(G\) is a _\(k\)-outer planar graph_, if it has a planar embedding such that removing the vertices of \(G\) incident to the outer face will result in a \((k-1)\)-outer planar graph. A \(k\)-outer planar graph \(G\) has a branchwidth of at most \(2k\). [PERSON]'s PTAS framework for a problem \(P\) in a planar graph \(G\) is to decompose \(G\) into a collection of \(k\)-outer planar subgraphs, find an optimal partial solution of \(P\) in each subgraph by an exact algorithm, and take the union of the optimal partial solutions as a solution of \(P\) in \(G\). When the framework is used for a minimization problem, \(G\) is decomposed in such a way that every two \"neighbor\" \(k\)-outer planar subgraphs share \"one-level\" vertices. [PERSON] shows that the framework gives a \(2^{O(k)}n\) time \((1+1/k)\)-approximation algorithm for the vertex cover problem in planar graphs and mentions that the framework can be applied to obtain \(2^{O(k)}n\) time \((1+1/k)\)-approximation algorithms for many other minimization problems, including the planar dominating set problem [9]. For a maximization problem like the independent set problem in planar graphs, [PERSON]'s framework gives a \(2^{O(k)}n\) time \(k/(k+1)\)-approximation algorithm. We show that the approximation ratio of [PERSON]'s framework is not bounded by any constant for the planar dominating set problem when two \"neighbor\" \(k\)-outer planar subgraphs share only \"one-level\" of vertices. To get a PTAS for the planar dominating set problem, the application of the framework has to be modified. We modify the application of the framework by decomposing \(G\) into \(k\)-outer planar subgraphs such that every two \"neighbor\" subgraphs share \"two-levels\" of vertices. Let \(O(2^{ck}n)\), \(c\) is a constant, be the running time of [PERSON]'s framework with \"one-level\" of overlapping vertices for the planar dominating set problem. We show that the modified application of the framework gives a PTAS with approximation ratio \((1+2/k)\) for the planar dominating set problem.