On generalized geometric graphs and pseudolines

This paper generalizes, extends and simplifies many results involving geometric graphs on planar point sets, in the context of pseudoline arrangements. Many of the problems studied here have been the focus of extensive research, and some of them are of central significance in combinatorial and computational geometry. Generalizations and simplifications of their solutions are therefore important in improving our understanding of these problems and in making them more accessible. (a) Using a duality transformation on pseudolines, established recently by Agarwal and Sharir [5], we show that any graph G induced by a set of vertices of an arrangement of a finite set of (x-monotone) pseudolines (referred to as a pseudoline graph) can be drawn in the plane such that its edges are `extendible pseudosegments' (in the terminology of [8]; see below), and such that two edges e 1 ; e 2 in G form a diamond (each of the two corresponding vertices lies above one pseudoline incident to the other vertex and below the other pseudoline) if and