Contact Representations of Directed Planar Graphs in 2D and 3D.

In applications such as VLSI floorplanning and cartogram design, vertices of a graph are represented by geometric objects and edges are captured by contacts between those objects, which are examples of a drawing style called contact graph representations. We study the feasibility of using line segments, triangles and tetrahedra to realize point-side contact representations for a number of graph classes including oriented versions of outerplanar graphs, 2-trees and 3-trees. Our main results show that every orientation of a maximal outerplanar graph of out-degree at most two, a 2-tree of out-degree at most two, and a planar 3-tree of out-degree at most four enjoy point-side contact representations using line segments, triangles, and tetrahedra, respectively. Unlike undirected graphs for which a fairly large amount of results can be found in the literature in the study of contact representations, directed graphs remain largely unexplored, and our study advances this line of research a step further.