Hanani-Tutte for Radial Planarity II

Abstract

Abstract. A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, . . . , C k with common center c, and edges are drawn radially: every edge intersects every circle centered at c 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 assignment of vertices to circles corresponds to the given 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 a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.

Strong Hanani-Tutte for the Torus

“…There is also work on applying Hanani-Tutte to different planarity variants, see, for example, [13,14,15,17,32]. A variant of the (strong) Hanani-Tutte theorem in the context of approximating maps of graphs, which works on any surface, was recently announced in a paper co-authored by the first author [7].…”

Level Planarity: Transitivity vs. Even Crossings

“…4 (a). Therefore, we introduce constraint (6). For each edge e in E + i ∪ E − i it remains to decide whether it is embedded locally to the left or to the right of ε i .…”

“…This results in a novel polynomial-time algorithm for testing radial level planarity by testing satisfiability of a system of constraints that, much like the work of Randerath et al, is obtained from omitting all transitivity constraints from a constraint system that trivially models radial level planarity. Currently, we deduce the correctness of the new algorithm from the strong Hanani-Tutte theorem for radial level planarity [6]. However, also this transformation works both ways, and a new correctness proof of our algorithm in…”

“…This was shown in the affirmative by Fulek et al [7], who also established the strong version for level planarity. Most recently, both the weak and the strong Hanani-Tutte theorem have been established for radial level planarity [5,6].…”

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Recently, Fulek et al [5,6,7] have presented Hanani-Tutte results for (radial) level planarity, i.e., a graph is (radial) level planar if it admits a (radial) level drawing where any two (independent) edges cross an even number of times. We show that the 2-Sat formulation of level planarity testing due to Randerath et al [14] is equivalent to the strong Hanani-Tutte theorem for level planarity [7].

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