Configurations of DNA cages based on plane graphs and vertex junctions

Motivated by recent advances in DNA nanotechnology, in this paper we study and provide answers to several general questions concerning the constructions of DNA cages. For example, if a DNA polyhedron can be constructed based on a given polyhedron in the form of a plane graph using certain vertex junction configurations, what is the minimum number of DNA strands needed, and how these vertex junction configurations can be chosen in order to achieve this minimum number? Using a new mathematical result obtained in this paper and the constructions used in its proof, we show explicitly that there is a way to choose the number and positions of certain junction configurations in the polyhedral graph so that the corresponding DNA cage (polyhedron) constructed will have the minimum number of DNA strands. (paper)