On representations of some thickness-two graphs - Extended abstract

This paper considers representations of graphs as rectangle-visibility graphs and as doubly linear graphs. These are, respectively, graphs whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical visibility, and graphs that can be drawn as the union of two straight-edged planar graphs. We prove that these graphs have, with n vertices, at most 6n-20 (resp., 6n-18) edges, and we provide examples of these graphs with 6n-20 edges for each n greater than or equal to 8.