Rectangle and Box Visibility Graphs in 3D

Rectangle-Visibility Representations of Bipartite Graphs

The paper considers representations of bipartite graphs as rectangle-visibility graphs, Le., graphs whose vertices are rectangles in the plane, with adjacency detenmned by horizontal and vertical visibility.1t is shown that, for p ~ q, Kp,q has a representation with no rectangles having collinear sides if and only if p ~ 2 or p = 3 and q ~ 4. More generally, it is shown that Kp;q is a rectangle...

On Visibility Graphs — Upper Bounds and Classification of Special Types

We examine several types of visibility graphs: bar and semi-bar k-visibility graphs, rectangle k-visibility graphs, arc and circle k-visibility graphs, and compact visibility graphs. We improve the upper bound on the thickness of bar k-visibility graphs from 2k(9k − 1) to 6k, and prove that the upper bound must be at least k + 1. We also show that the upper bound on the thickness of semi-bar k-...

More directions in visibility graphs

In this paper we introduce unit bar k-visibility graphs, which are bar kvisibility graphs in which every bar has unit length. We show that almost all families of unit bar k-visibility graphs and unit bar k-visibility graphs are incomparable under set inclusion. In addition, we establish the largest complete graph that is a unit bar k-visibility graph. As well, we present a family of hyperbox vi...

On Rectangle Visibility Graphs. III. External Visibility and Complexity

Rectangle-visibility Layouts of Unions and Products of Trees

The paper considers representations of unions and products of trees as rectangle-visibility graphs (abbreviated RVGs), i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. Our main results are that the union of any tree (or forest) with a depth-1 tree is an RVG, and that the union of two depth-2 trees and the union of a depth-...