Abstract
It is shown that every maximal planar graph (triangulation) can be contracted at an arbitrary point (by identifying it with an adjacent point) so that triangularity is preserved. This is used as a lemma to prove that every triangulation can be (a) oriented so that with three exceptions every point has indegree three, the exceptions having indegrees 0, 1 and 2, and (b) decomposed into three edge-disjoint trees of equal order. The lemma also provides an elementary proof of a theorem of Wagner that every triangulation can be represented by a straight-line drawing.
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