## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, September 22, 2016, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Lazar Todorovic

## Size of Separators in GIRGs Depends on Underlying Geometry

This talk gives a brief overview of my recently completed Master Thesis. In Euclidean Geometric Inhomogeneous Random Graphs (GIRG), a model for real-networks recently presented by Bringmann, Keusch, and Lengler, each vertex is equipped with a weight and a position. The weights are drawn from a power-law distribution (with a fixed exponent $2 < \beta < 3$), while the positions are drawn uniformly from the $d$-dimensional torus $\mathbb{T}^d$. Then, the edge probability of two vertices is roughly polynomial in the product of their weights, and the inverse of their Euclidean distance. The Euclidean GIRG model has a wide range of interesting properties, such as a small diameter, small average distance, small separators, and a large clustering coefficient. The focus of the thesis lies on studying a variant of the GIRG model that uses minimum component distance (MCD) instead of Euclidean distance, but is otherwise defined in the same way as its Euclidean counterpart. The main result of the thesis is that the MCD model (for $d>1$) has no small (edge) separators, setting it apart from Euclidean GIRGs, and showing that the underlying geometry has a crucial influence on the separator property of GIRGs.

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