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. 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, 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 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.
Research by J. Kratochvíl and P. Valtr was supported by the Czech Science Foundation (GAČR) grant no. 18-19158S. Research by F. Montecchiani partially supported by: (i) MIUR, under Grant 20174LF3T8 AHeAD: efficient Algorithms for HArnessing networked Data. (ii) Dipartimento di Ingegneria dell’Università degli Studi di Perugia, under grant RICBA18WD: “Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni”.
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Acknowledgments
Research started at the 2017 Bertinoro Workshop on Graph Drawing, we thank all participants for fruitful discussions.
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Kaufmann, M., Kratochvil, J., Lipp, F., Montecchiani, F., Raftopoulou, C., Valtr, P. (2020). The Stub Resolution of 1-Planar Graphs. In: Rahman, M., Sadakane, K., Sung, WK. (eds) WALCOM: Algorithms and Computation. WALCOM 2020. Lecture Notes in Computer Science(), vol 12049. Springer, Cham. https://doi.org/10.1007/978-3-030-39881-1_15
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