Visibility Drawings of Graphs on Surfaces

After showing that any graph admits a visibility bar-representation in an adequate closed surface, we establish the basis to the study of the relationship between Visibility and Topological representations of graphs in the plane and in other surfaces. Thus, we present an algorithm that, in linear time, bar-represents trees on the surface of less genus possible. Keywords: Visibility, drawing graphs, bar-visibility 1 Introduction In the most usual way to represent graphs (topological representation) vertices map to points in the space or the plane and edges are represented by Jordan curves joining those points. In this way, a combinatorial structure as a graph has been represented typically by a topological space or, more concretely, by a subspace (the underlying space) of the Euclidean space or the Euclidean plane. Although this representation is very convenient from many points of view, it forgets the combinatorial nature of the graphs and infinitely many graphs share the same underl..