On rectilinear drawing of graphs

A rectilinear drawing is an orthogonal grid drawing without bends, possibly with edge crossings, without any overlapping between edges, between vertices, or between edges and vertices. Rectilinear drawings without edge crossings (planar rectilinear drawings) have been extensively investigated in graph drawing. Testing rectilinear planarity of a graph is NP-complete [10]. Restricted cases of the planar rectilinear drawing problem, sometimes called the "no-bend orthogonal drawing problem", have been well studied (see, for example,[13],[14],[15] ). In this paper, we study the problem of general non-planar rectilinear drawing; this problem has not received as much attention as the planar case. We consider a number of restricted classes of graphs and obtain a polynomial time algorithm, NP-hardness results, an FPT algorithm, and some bounds. We define a structure called a "4-cycle block". We give a linear time algorithm to test whether a graph that consists of a single 4-cycle block has a rectilinear drawing, and draw it if such a drawing exists. We show that the problem is NP-hard for the graphs that consist of 4-cycle blocks connected by single edges, as well as the case where each vertex has degree 2 or 4. We present a linear time fixed-parameter tractable algorithm to test whether a degree-4 graph has a rectilinear drawing, where the parameter is the number of degree-3 and degree-4 vertices of the graph. We also present a lower bound on the area of rectilinear drawings, and a upper bound on the number of edges.