Boris Horvat

Products of unit distance graphs

Some graphs admit drawings in the Euclidean plane (k-space) in such a (natural) way, that edges are represented as line segments of unit length. We say that they have the unit distance property. The influence of graph operations on the unit distance property is discussed. It is proved that the Cartesian product preserves the unit distance property in the Euclidean plane, while graph union, join, tensor product, strong product, lexicographic product and corona do not. It is proved that the Cartesian product preserves the unit distance property also in higher dimensions.

On the computational complexity of degenerate unit distance representations of graphs

Some graphs admit drawings in the Euclidean k-space in such a (natural) way, that edges are represented as line segments of unit length. Such embeddings are called k-dimensional unit distance representations. The embedding is strict if the distances of points representing nonadjacent pairs of vertices are different than 1.When two nonadjacent vertices are drawn in the same point, we say that the representation is degenerate. Computational complexity of nondegenerate embeddings has been studied before. We initiate the study of the computational complexity of (possibly) degenerate embeddings. In particular we prove that for every k≥ 2, deciding if an input graph has a (possibly) degenerate k-dimensional unit distance representation is NP-hard.

Baselines for the Preparation of Electronic Textbooks

Baselines for the Preparation of Electronic Textbooks The NAUK group (Advanced Learning Blocks group; http://www.nauk.si) is engaged in the development of theoretical and practical concepts of ICT use in all levels of education. Recently, the proposed introduction of electronic textbooks (e-textbooks) has become a major topic nationwide in Slovenia. There are multiple dilemmas to be solved, e.g. what a modern e-textbook is, what it should include, in what way should it differ from conventional textbooks, what tools and technologies should be used for its creation, etc. This paper attempts to make recommendations, which we believe should be followed by all the authors of e-textbooks.