Bounded Stub Resolution for Some Maximal 1-Planar Graphs

Abstract

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 $\frac{1}{5}$.

This research was initiated at the Bertinoro Workshop on Graph Drawing 2017. Research by J. Kratochvíl and P. Valtr was supported by project CE-ITI no. P202/12/G061 of the Czech Science Foundation (GAČR). F. Lipp was partially supported by Cusanuswerk. Research of Fabrizio Montecchiani supported in part by the project: “Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni”- Ricerca di Base 2017, Dipartimento di Ingegneria dell’Universita degli Studi di Perugia”.