The absence of efficient dual pairs of spanning trees in planar graphs (Q2501002)

Summary: 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.