@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{10773,
  abstract     = {The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.},
  author       = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  pages        = {811--842},
  publisher    = {Springer Nature},
  title        = {{Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics}},
  doi          = {10.1007/s00454-022-00371-2},
  volume       = {67},
  year         = {2022},
}

@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},
}

@article{11660,
  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 collections of interrelated sorted lists together with their persistence diagrams. },
  author       = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  journal      = {LIPIcs},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs}},
  year         = {2022},
}

@inproceedings{9604,
  abstract     = {Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.},
  author       = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  booktitle    = {Leibniz International Proceedings in Informatics},
  isbn         = {9783959771849},
  issn         = {18688969},
  location     = {Online},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory}},
  doi          = {10.4230/LIPIcs.SoCG.2021.16},
  volume       = {189},
  year         = {2021},
}