@inproceedings{15012,
  abstract     = {We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n-1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.},
  author       = {Pach, János and Saghafian, Morteza and Schnider, Patrick},
  booktitle    = {31st International Symposium on Graph Drawing and Network Visualization},
  isbn         = {9783031492716},
  issn         = {16113349},
  location     = {Isola delle Femmine, Palermo, Italy},
  pages        = {339--346},
  publisher    = {Springer Nature},
  title        = {{Decomposition of geometric graphs into star-forests}},
  doi          = {10.1007/978-3-031-49272-3_23},
  volume       = {14465},
  year         = {2024},
}

@article{14345,
  abstract     = {For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).},
  author       = {Edelsbrunner, Herbert and Garber, Alexey and Ghafari, Mohadese and Heiss, Teresa and Saghafian, Morteza},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  publisher    = {Springer Nature},
  title        = {{On angles in higher order Brillouin tessellations and related tilings in the plane}},
  doi          = {10.1007/s00454-023-00566-1},
  year         = {2023},
}

@article{13182,
  abstract     = {We characterize critical points of 1-dimensional maps paired in persistent homology
geometrically and this way get elementary proofs of theorems about the symmetry
of persistence diagrams and the variation of such maps. In particular, we identify
branching points and endpoints of networks as the sole source of asymmetry and
relate the cycle basis in persistent homology with a version of the stable marriage
problem. Our analysis provides the foundations of fast algorithms for maintaining a
collection of sorted lists together with its persistence diagram.},
  author       = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  issn         = {2367-1734},
  journal      = {Journal of Applied and Computational Topology},
  publisher    = {Springer Nature},
  title        = {{Geometric characterization of the persistence of 1D maps}},
  doi          = {10.1007/s41468-023-00126-9},
  year         = {2023},
}

@article{11658,
  abstract     = {The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.},
  author       = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  journal      = {Leibniz International Proceedings on Mathematics},
  publisher    = {Schloss Dagstuhl - Leibniz Zentrum für Informatik},
  title        = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}},
  year         = {2022},
}