TY - CONF AB - A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a probabilistic FPT-approach in the number of popular faces. AU - De Nooijer, Phoebe AU - Terziadis, Soeren AU - Weinberger, Alexandra AU - Masárová, Zuzana AU - Mchedlidze, Tamara AU - Löffler, Maarten AU - Rote, Günter ID - 14888 SN - 0302-9743 T2 - 31st International Symposium on Graph Drawing and Network Visualization TI - Removing popular faces in curve arrangements VL - 14466 ER - TY - CONF AB - 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. AU - Pach, János AU - Saghafian, Morteza AU - Schnider, Patrick ID - 15012 SN - 03029743 T2 - 31st International Symposium on Graph Drawing and Network Visualization TI - Decomposition of geometric graphs into star-forests VL - 14465 ER - TY - THES AB - Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in discrete geometry that have captivated mathematicians for centuries, if not millennia. This thesis seeks to cast new light on these structures by illustrating specific instances where a topological perspective, specifically through discrete Morse theory and persistent homology, provides valuable insights. At first glance, the topology of these geometric objects might seem uneventful: point sets essentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which is a contractible space, and the topology of a network primarily involves the enumeration of connected components and cycles within the network. However, beneath this apparent simplicity, there lies an array of intriguing structures, a small subset of which will be uncovered in this thesis. Focused on three case studies, each addressing one of the mentioned objects, this work will showcase connections that intertwine topology with diverse fields such as combinatorial geometry, algorithms and data structures, and emerging applications like spatial biology. AU - Cultrera di Montesano, Sebastiano ID - 15094 SN - 2663 - 337X TI - Persistence and Morse theory for discrete geometric structures ER - TY - CONF AB - We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right. AU - Cultrera di Montesano, Sebastiano AU - Edelsbrunner, Herbert AU - Henzinger, Monika H AU - Ost, Lara ED - Woodruff, David P. ID - 15093 T2 - Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) TI - Dynamically maintaining the persistent homology of time series ER - TY - GEN AB - 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. AU - Cultrera di Montesano, Sebastiano AU - Draganov, Ondrej AU - Edelsbrunner, Herbert AU - Saghafian, Morteza ID - 15091 T2 - arXiv TI - Chromatic alpha complexes ER -