Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, November 12, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Michael Hoffmann

On Universal Geometric Graphs

A graph G is universal for a class H of graphs if G contains every graph from H as a subgraph. Obviously, the complete graph K_n is universal for all graphs on n vertices, and it is the only graph with this property. Things get more interesting if we restrict our focus to a subclass C of n-vertex graphs and ask what is the minimum number of edges in an n-vertex graph that is universal for C. This type of question is well studied in the abstract setting. We study it in a geometric setting for families of planar graphs and plane drawings. Specifically, we obtain an n-vertex geometric graph with O(n log n) edges that is universal for plane forests on n vertices. The construction is based on a classic result by Chung and Graham for the abstract setting, and the bound on the number of edges is asymptotically optimal. Furthermore, we study the number of edges required in a convex geometric graph on n vertices that is universal for n-vertex outerplanar graphs. (Joint work with Fabrizio Frati and Csaba D. Tóth.)

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