**O. Aichholzer, F. Aurenhammer, and R. Hainz**

Let be a polygon in the plane. We disprove the conjecture that the
so-called LMT-skeleton coincides with the intersection of all locally minimal
triangulations, , even for convex polygons . We introduce an
improved LMT-skeleton algorithm which, for any simple polygon , exactly
computes , and thus a larger subgraph of the minimum-weight
triangulation . The algorithm achieves the same in the general point
set case provided the connectedness of the improved LMT-skeleton, which is
given in allmost all practical instances. We further observe that the
-skeleton of is a subset of for all values
provided is convex or near-convex. This gives
evidence for the tightness of this bound in the general point set case.