Abstrakt
The resolution of a drawing plays a crucial role when defining criteria for its quality and readability. In the past, grid resolution, edge-length resolution, angular resolution and crossing resolution have been investigated.
We continue the study of the recently introduced stub resolution as an additional aesthetic criterion for nonplanar drawings of graphs. A crossed edge is divided into parts, called stubs, which should not be too short for the sake of readability.
Thus, the stub resolution of a drawing is defined as the minimum ratio between the length of a stub and the length of the entire edge containing that stub, over all the edges of the drawing. As a meaningful graph class, where crossings are naturally involved, we consider 1-planar graphs (i.e., graphs that allow planar drawings in which every edge is crossed at most once).
In an attempt to prove the conjecture that the stub resolution of 1-planar graphs is bounded, we closely investigate a class of maximal 1-planar graphs arising from double-wheels. We show that each such graph allows a straight-line 1-planar drawing with stub resolution 1/5.