Geometry Seminar
Monotone Arc Diagrams with Few Biarcs
Speaker: Michael Hoffmann, ETH Zürich
Location: Online Zoom-only
Videoconference link: https://youtu.be/HlE4E-2NB0g
Date: Tuesday, November 19, 2024, 2 p.m.
Synopsis:
Arc diagrams are plane embeddings of graphs that represent vertices as points on a horizontal line, called spine, and each edge as a sequence of halfcircles centered on the spine. If each such sequence consists of one halfcircle only, then we face a 2-page book embedding. Not every planar graph admits a 2-page book embedding, only the subgraphs of Hamiltonian planar graphs do. But every planar graph admits an arc diagram where each edge is represented as a sequence of two halfcircles, a so-called biarc. Furthermore, we can restrict all biarcs to be monotone with respect to the spine. Some number of edges can always be represented as proper arcs, that is, as a single halfcircle. But how many? For a planar graph G, let mhd(G) denote the minimum number of biarcs over all arc diagrams of G where each edge is represented as a proper arc or as a monotone biarc. In this talk I will discuss upper and lower bounds on mhd for some classes of planar graphs.
Based on joint work with Steven Chaplick, Henry Förster, and Michael Kaufmann that appeared at GD this year.Notes:
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