Triangulations without minimum-weight drawing

It is known that some triangulation graphs admit straight-line drawings rea lizing certain characteristics, e.g., greedy triangulation, minimum-weight triangulation, Delaunay triangulation, etc. Lenhart and Liotta (1997) in th eir pioneering paper on "drawable" minimum-weight triangulations raised an open problem: 'Does every triangulation graph whose skeleton is a forest ad mit a minimum-weight drawing?' In this paper, we answer this problem by dis proving it in the general case and even when the skeleton is restricted to a tree or, in particular, a star. (C) 2000 Elsevier Science B.V. All rights reserved.