## 1 Introduction  
An important research area in graph theory and networks is domination; it has been energetically investigated for many years due to its large number of real-world applications, such as resource allocation [1,2] and voting [3]. [PERSON] _et al_. In their books, [4,5] provide a good survey on domination problems. Let \(G\) be a simple undirected graph with the set of vertices \(V(G)\) and the set of edges \(E(G)\). We denote \(|V(G)|\) by \(n\). The \(r\)-dominating set \(D\) of \(G\) is a subset of \(V(G)\) containing \(r\) vertices, such that for every vertex \(v\) in \(V(G)\), either \(v\in D\) or \(v\) is adjacent to a vertex in \(D\). The minimum integer \(r\) for which \(G\) has a \(r\)-dominating set is called the _domination number_ of \(G\) and is denoted by \(\gamma(G)\). The dominating set problem is to decide that given a graph \(G\) and an integer \(r\), whether \(\gamma(G)\leq r\). The optimization version of this problem is to find a minimum dominating set. The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory [6]. There is a long history of research on the approximation and exact algorithms to tackle the intractability of the problem. A minimization problem \(P\) is \(\alpha-\)approximable \((\alpha\geq 1)\) if there is an algorithm which gives a solution for any instance of \(P\) in polynomial time in the instance size with solution value at most \(\alpha OPT\), where \(OPT\) is the value of an optimal solution for the instance of \(P\). If \(P\) is \((1+\epsilon)-\)approximable for any fixed \(\epsilon>0\) then \(P\) has a polynomial time approximation scheme (PTAS). The dominating set problem for general graphs is \((1+\log n)-\)approximable [7], however, it is not approximable within a factor \((1-\epsilon)\ln n\) for any \(\epsilon>0\) unless \(NP\subseteq DTIME(n^{\log\log n})\)[8]. The dominating set problem has been widely studied on an important class of graphs, the _planar_ graphs. A graph is planar if it can be drawn on the sphere with no crossing edges. The dominating set problem in planar graphs (planar dominating set problem) remains NP-hard [6] but admits a PTAS [9]. The fixed parameter algorithms have played a central role in exact algorithms for the planar dominating set problem. A minimization problem \(P\) is fixed-parameter tractable if given a parameter \(r\), whether \(OPT\) of \(P\) is at most \(r\) can be decided in \(f(r)n^{O(1)}\) time, where \(f(r)\) is a computable function depending only on \(r\)[10]. Such an algorithm is called a _fixed parameter tractable_ (FPT) algorithm. Readers may refer to [11] for a survey on new techniques for developing exact algorithms for NP-hard problems. It is shown in [10] that for general graphs, the dominating set problem is not fixed-parameter tractable unless some collapses occur between parametrized complexity classes. However, the planar dominating set problem is fixed-parameter tractable [10]. The planar dominating set problem also admits a linear size kernel [12]. Recent progresses in FPT algorithms result in subexponential time exact algorithms for the planar dominating set problem [13, 14, 15]. These algorithms use the tree-/branch-decomposition based approach and have running time \(O(2^{c\sqrt{\gamma(G)}}n+n^{O(1)})\), \(c\) is a constant. The branch-decomposition based algorithm by [PERSON] (called FT algorithm in what follows) [14] achieves a smallest constant \(c\) in the exponent of the running time. The notion of branch-decomposition of graphs is introduced by [PERSON] and [PERSON] [16]. Informally, a branch-decomposition of a graph \(G\) is a collection of vertex cut sets of \(G\) that decomposes \(G\) into subgraphs with each edge of \(G\) a minimal subgraph. The width of a branch-decomposition is the maximum size of the vertex cuts in the collection. The branchwidth of \(G\), denoted by \(\mathrm{bw}(G)\), is the minimum width of all possible branch-decompositions of \(G\).