The Stub Resolution of 1-planar Graphs

Abstract

The resolution of a drawing plays a crucial role when defining criteria for its quality. In the past, grid resolution, edge-length resolution, angular resolution and crossing resolution have been investigated. In this paper, we investigate the stub resolution, a recently introduced criterion for nonplanar drawings. Intersection points divide edges 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, over all the edges of the drawing. We consider $1$-planar graphs and we explore scenarios in which near optimal stub resolution, i.e., arbitrarily close to $\frac{1}{2}$, can be obtained in drawings with zero, one or two bends per edge, as well as further resolution criteria, such as angular and crossing resolution. In particular, our main contributions are as follows: (i) Every IC-planar graph, i.e., every $1$-planar graph with independent crossing edges, has a straight-line drawing with near optimal stub resolution; (ii) Every $1$-planar graph has a $1$-bend drawing with near optimal stub resolution.