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{\bf T.R. Riley and W.P. Thurston}
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{\bf The Absence of Efficient Dual Pairs of Spanning Trees in Planar Graphs}
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A spanning tree $T$ in a finite planar connected graph $G$ determines
a dual spanning tree $T\*$ in the dual graph $G\*$ such that $T$ and
$T\*$ do not intersect. We show that it is not always possible to
find $T$ in $G$ such that the diameters of $T$ and $T\*$ are both
within a uniform multiplicative constant (independent of $G$) of the
diameters of their ambient graphs.
\bye