@inproceedings{21374, abstract = {Let . S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on . S is a crossing-free geometric path such that every point of .S is a vertex of the path. It is known that, if. S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on . S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path . π with endpoints .s, t ∈ S and constraints given by a segment . ab, where .a, b ∈ S. We consider the following scenarios: (i) .ab ∈ π; (ii) .ab π. We characterize those quintuples . S, a, b, s, t for which . π exists. Secondly, we consider the problem of finding two plane Hamiltonian paths . π1, π2 on a set . S with constraints given by a segment . ab, where .a, b ∈ S. We consider the following scenarios: (i) .π1 and .π2 share no edges and .ab is an edge of . π1; (ii) .π1 and .π2 share no edges and none of them includes .ab as an edge; (iii) both .π1 and .π2 include .ab as an edge and share no other edges. In all cases, we characterize those triples . S, a, b for which .π1 and .π2 exist.}, author = {Antić, Todor and Džuklevski, Aleksa and Fiala, Jiří and Kratochvíl, Jan and Liotta, Giuseppe and Saghafian, Morteza and Saumell, Maria and Zink, Johannes}, booktitle = {51st International Conference on Current Trends in Theory and Practice of Computer Science}, isbn = {9783032178008}, issn = {1611-3349}, location = {Krakow, Poland}, pages = {532--546}, publisher = {Springer Nature}, title = {{Edge-constrained Hamiltonian paths on a point set}}, doi = {10.1007/978-3-032-17801-5_39}, volume = {16448}, year = {2026}, } @inproceedings{21410, abstract = {Given a finite set of red and blue points in R^d, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The MST-ratio has recently gained attention due to its direct interpretation in topological models for studying point sets with applications in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio over all proper colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time 3-approximation algorithm for this problem. As part of the proof, we show that in any metric space, the maximum MST-ratio is smaller than 3. Furthermore, we study the average MST-ratio over all colorings of a set of n points. We show that this average is always at least n-2/n-1, and for n random points uniformly distributed in a d-dimensional unit cube, the average tends to (math formular) in expectation as n approaches infinity.}, author = {Jabal Ameli, Afrouz and Motiei, Faezeh and Saghafian, Morteza}, booktitle = {20th International Conference and Workshops on Algorithms and Computation}, isbn = {9789819571260}, issn = {1611-3349}, location = {Perugia, Italy}, pages = {386--401}, publisher = {Springer Nature}, title = {{On the MST-ratio: Theoretical bounds and complexity of finding the maximum}}, doi = {10.1007/978-981-95-7127-7_26}, volume = {16444}, year = {2026}, } @article{21781, abstract = {Given a set A of n points (vertices) in general position in the plane, the complete geometric graph Kn[A] consists of all (n2) segments (edges) between the elements of A. It is known that the edge set of every complete geometric graph on n vertices can be partitioned into O(n3∕2) crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set A of n randomly selected points, uniformly distributed in [0,1]2, with probability tending to 1 as n→∞, the edge set of Kn[A] can be covered by O(nlogn) crossing-free paths and by O(n√logn) crossing-free matchings. On the other hand, we construct n-element point sets such that covering the edge set of Kn[A] requires a quadratic number of monotone paths.}, author = {Dumitrescu, Adrian and Pach, János and Saghafian, Morteza and Scott, Alex}, issn = {2996-220X}, journal = {Combinatorics and Number Theory}, number = {1}, pages = {73--82}, publisher = {Mathematical Sciences Publishers}, title = {{Covering complete geometric graphs by monotone paths}}, doi = {10.2140/cnt.2026.15.73}, volume = {15}, year = {2026}, } @article{20456, abstract = {Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {24--47}, publisher = {Springer Nature}, title = {{On the size of chromatic Delaunay mosaics}}, doi = {10.1007/s00454-025-00778-7}, volume = {75}, year = {2026}, } @article{20490, abstract = {We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips. }, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafari, Mohadese and Heiss, Teresa and Saghafian, Morteza}, issn = {0195-6698}, journal = {European Journal of Combinatorics}, publisher = {Elsevier}, title = {{Flips in two-dimensional hypertriangulations}}, doi = {10.1016/j.ejc.2025.104248}, volume = {132}, year = {2026}, } @article{20585, abstract = {Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, issn = {2639-8001}, journal = {Foundations of Data Science}, keywords = {Topological data analysis, Delaunay mosaic, alpha complex, chromatic sets, persistent homology, kernel/image/cokernel persistent homology, radius function, discrete Morse theory, exact sequences}, pages = {30--62}, publisher = {AIMS}, title = {{Chromatic alpha complexes}}, doi = {10.3934/fods.2025003}, volume = {8}, year = {2026}, } @article{21253, 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}, issn = {0925-7721}, journal = {Computational Geometry}, publisher = {Elsevier}, title = {{Decomposition of geometric graphs into star-forests}}, doi = {10.1016/j.comgeo.2025.102186}, volume = {129}, year = {2025}, } @article{18626, abstract = {The local angle property of the (order-1) Delaunay triangulations of a generic set in R2 asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.}, author = {Edelsbrunner, Herbert and Garber, Alexey and Saghafian, Morteza}, issn = {1090-2082}, journal = {Advances in Mathematics}, publisher = {Elsevier}, title = {{Order-2 Delaunay triangulations optimize angles}}, doi = {10.1016/j.aim.2024.110055}, volume = {461}, year = {2025}, } @article{19937, abstract = {Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study.}, author = {Mahini, Mohammad and Beigy, Hamid and Qadami, Salman and Saghafian, Morteza}, issn = {0020-0255}, journal = {Information Sciences}, number = {11}, publisher = {Elsevier}, title = {{Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality}}, doi = {10.1016/j.ins.2025.122425}, volume = {719}, year = {2025}, } @inproceedings{20005, abstract = {We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.}, author = {Edelsbrunner, Herbert and Garber, Alexey and Saghafian, Morteza}, booktitle = {41st International Symposium on Computational Geometry}, isbn = {9783959773706}, issn = {1868-8969}, location = {Kanazawa, Japan}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{On spheres with k points inside}}, doi = {10.4230/LIPIcs.SoCG.2025.43}, volume = {332}, year = {2025}, } @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}, pages = {29--48}, publisher = {Springer Nature}, title = {{On angles in higher order Brillouin tessellations and related tilings in the plane}}, doi = {10.1007/s00454-023-00566-1}, volume = {72}, year = {2024}, } @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 = {1611-3349}, 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{15380, 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}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, pages = {557--578}, publisher = {Springer Nature}, title = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}}, doi = {10.1007/s41468-024-00173-w}, volume = {8}, year = {2024}, } @inproceedings{17145, abstract = {Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.}, author = {Rote, Günter and Rüber, Moritz and Saghafian, Morteza}, booktitle = {40th International Symposium on Computational Geometry}, isbn = {9783959773164}, issn = {1868-8969}, location = {Athens, Greece}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Grid peeling of parabolas}}, doi = {10.4230/LIPIcs.SoCG.2024.76}, volume = {293}, year = {2024}, } @inproceedings{18556, abstract = {Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization}, isbn = {9783959773430}, issn = {1868-8969}, location = {Vienna, Austria}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The Euclidean MST-ratio for bi-colored lattices}}, doi = {10.4230/LIPIcs.GD.2024.3}, volume = {320}, year = {2024}, } @unpublished{18999, abstract = {Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{Chromatic topological data analysis}}, doi = {10.48550/ARXIV.2406.04102}, year = {2024}, } @unpublished{15091, abstract = {Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{Chromatic alpha complexes}}, doi = {10.48550/arXiv.2212.03128}, year = {2024}, } @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}, pages = {1101--1119}, publisher = {Springer Nature}, title = {{Geometric characterization of the persistence of 1D maps}}, doi = {10.1007/s41468-023-00126-9}, volume = {8}, year = {2024}, } @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}, } @unpublished{15090, abstract = {Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.}, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{On the size of chromatic Delaunay mosaics}}, year = {2022}, }