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Abstract

Let $G$ be a biconnected planar graph given together with its planar drawing. A {\em face-vertex walk} in $G$ of length $k$ is an alternating sequence $x_0, \ldots x_k$ of vertices and faces (i.e., if $x_{i - 1}$ is a face then $x_i$ is a vertex and vice versa) such that $x_{i - 1}$ and $x_i$ are incident with each other for $1 \leq i \leq k$. For each vertex or face $x$ of $G$, let $\alpha_x$ denote the length of the shortest face-vertex walk from the outer face of $G$ to $x$. Let $\alpha_G$ denote the maximum of $\alpha_x$ over all vertices/faces $x$. We show that there always exits a branch-decomposition of $G$ with width $\alpha_G$ and that such a decomposition can be constructed in linear time. We also give experimental results, in which we compare the width of our decomposition with the optimal width and with the width obtained by some heuristics for general graphs proposed by previous researchers, on test instances used by those researchers. On 56 out of the total 59 test instances, our method gives a decomposition within additive 2 of the optimum width and on 33 instances it achieves the optimum width.