Triangulations without Minimum-Weight Drawing

It is known that some triangulation graphs admit straight-line drawings realizing certain characteristics, e.g., greedy triangulation, minimumweight triangulation, Delaunay triangulation, etc.. Lenhart and Liotta [12] in their pioneering paper on "drawable" minimum-weight triangulations raised an open problem: 'Does every triangulation graph whose skeleton is a forest admit a minimum-weight drawing?' In this paper, we answer this problem by disproving it in the general case and even when the skeleton is restricted to a tree or, in particular, a star.

[1]  Clyde L. Monma,et al.  Transitions in geometric minimum spanning trees (extended abstract) , 1991, SCG '91.

[2]  Robert J. Cimikowski.  Properties of some Euclidean proximity graphs , 1992, Pattern Recognit. Lett..

[3]  J. Mark Keil,et al.  Computing a Subgraph of the Minimum Weight Triangulation , 1994, Comput. Geom..

[4]  Clyde L. Monma,et al.  Transitions in geometric minimum spanning trees , 1991, SCG '91.

[5]  Peter Eades,et al.  The realization problem for Euclidean minimum spanning trees is NP-hard , 1994, SCG '94.

[6]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[7]  Giuseppe Liotta,et al.  Drawing Outerplanar Minimum Weight Triangulations , 1996, Inf. Process. Lett..

[8]  Boting Yang,et al.  Maximum Weight Triangulation and Graph Drawing , 1999, Inf. Process. Lett..

[9]  Michael B. Dillencourt.  Toughness and Delaunay triangulations , 1990, Discret. Comput. Geom..

[10]  Giuseppe Liotta,et al.  Drawable and Forbidden Minimum Weight Triangulations , 1997, Graph Drawing.

[11]  Giuseppe Liotta,et al.  Proximity Drawability: a Survey , 1994, Graph Drawing.

[12]  Yin-Feng Xu,et al.  A New Subgraph of Minimum Weight Triangulations , 1997, J. Comb. Optim..

[13]  Giuseppe Liotta,et al.  Recognizing Rectangle of Influence Drawable Graphs , 1994, Graph Drawing.

[14]  Prosenjit Bose,et al.  Proximity Constraints and Representable Trees , 1994, Graph Drawing.

[15]  D. Matula,et al.  Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane , 2010 .

[16]  Ioannis G. Tollis,et al.  Algorithms for Drawing Graphs: an Annotated Bibliography , 1988, Comput. Geom..