A drawing of a graph $G$, possibly with multiple edges but without loops, is radial if all edges are drawn radially, that is, each edge intersects every circle centered at the origin at most once. $G$ is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of $G$ are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the distances of the vertices from the origin respect the ordering or leveling. A pair of edges $e$ and $f$ in a graph is independent if $e$ and $f$ do not share a vertex. We show that if a leveled graph $G$ has a radial drawing in which every two independent edges cross an even number of times, then $G$ is radial planar. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.
@article{10_37236_10169,
author = {Radoslav Fulek and Michael Pelsmajer and Marcus Schaefer},
title = {Hanani-Tutte for radial planarity. {II}},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/10169},
zbl = {1507.05064},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10169/}
}
TY - JOUR
AU - Radoslav Fulek
AU - Michael Pelsmajer
AU - Marcus Schaefer
TI - Hanani-Tutte for radial planarity. II
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10169/
DO - 10.37236/10169
ID - 10_37236_10169
ER -
%0 Journal Article
%A Radoslav Fulek
%A Michael Pelsmajer
%A Marcus Schaefer
%T Hanani-Tutte for radial planarity. II
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10169/
%R 10.37236/10169
%F 10_37236_10169
Radoslav Fulek; Michael Pelsmajer; Marcus Schaefer. Hanani-Tutte for radial planarity. II. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/10169
