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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.
