@inproceedings{7994, abstract = {In the recent study of crossing numbers, drawings of graphs that can be extended to an arrangement of pseudolines (pseudolinear drawings) have played an important role as they are a natural combinatorial extension of rectilinear (or straight-line) drawings. A characterization of the pseudolinear drawings of K_n was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.}, author = {Arroyo Guevara, Alan M and Bensmail, Julien and Bruce Richter, R.}, booktitle = {36th International Symposium on Computational Geometry}, isbn = {9783959771436}, issn = {18688969}, location = {Zürich, Switzerland}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Extending drawings of graphs to arrangements of pseudolines}}, doi = {10.4230/LIPIcs.SoCG.2020.9}, volume = {164}, year = {2020}, } @inproceedings{7990, abstract = {Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on P is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge, removes a non-extreme point of degree 3, or adds a point in P ⧵ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The goal of this paper is to investigate the structure of this graph, with emphasis on its connectivity. For sets P of n points in general position, we show that the bistellar flip graph is (n-3)-connected, thereby answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos) on triangulations. This matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points and projecting the lower convex hull), where (n-3)-connectivity has been known since the late 1980s through the secondary polytope (Gelfand, Kapranov, Zelevinsky) and Balinski’s Theorem. Our methods also yield the following results (see the full version [Wagner and Welzl, 2020]): (i) The bistellar flip graph can be covered by graphs of polytopes of dimension n-3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n-3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations are regular iff the trivial subdivision has height n-3 in the partial order of partial subdivisions. (iv) There are arbitrarily large sets P with non-regular partial triangulations, while every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular partial triangulations (answering a question by F. Santos in the unexpected direction).}, author = {Wagner, Uli and Welzl, Emo}, booktitle = {36th International Symposium on Computational Geometry}, isbn = {9783959771436}, issn = {18688969}, location = {Zürich, Switzerland}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Connectivity of triangulation flip graphs in the plane (Part II: Bistellar flips)}}, doi = {10.4230/LIPIcs.SoCG.2020.67}, volume = {164}, year = {2020}, } @inproceedings{7807, abstract = {In a straight-line embedded triangulation of a point set P in the plane, removing an inner edge and—provided the resulting quadrilateral is convex—adding the other diagonal is called an edge flip. The (edge) flip graph has all triangulations as vertices, and a pair of triangulations is adjacent if they can be obtained from each other by an edge flip. The goal of this paper is to contribute to a better understanding of the flip graph, with an emphasis on its connectivity. For sets in general position, it is known that every triangulation allows at least edge flips (a tight bound) which gives the minimum degree of any flip graph for n points. We show that for every point set P in general position, the flip graph is at least -vertex connected. Somewhat more strongly, we show that the vertex connectivity equals the minimum degree occurring in the flip graph, i.e. the minimum number of flippable edges in any triangulation of P, provided P is large enough. Finally, we exhibit some of the geometry of the flip graph by showing that the flip graph can be covered by 1-skeletons of polytopes of dimension (products of associahedra). A corresponding result ((n – 3)-vertex connectedness) can be shown for the bistellar flip graph of partial triangulations, i.e. the set of all triangulations of subsets of P which contain all extreme points of P. This will be treated separately in a second part.}, author = {Wagner, Uli and Welzl, Emo}, booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms}, isbn = {9781611975994}, location = {Salt Lake City, UT, United States}, pages = {2823--2841}, publisher = {SIAM}, title = {{Connectivity of triangulation flip graphs in the plane (Part I: Edge flips)}}, doi = {10.1137/1.9781611975994.172}, volume = {2020-January}, year = {2020}, } @article{9308, author = {Avvakumov, Sergey and Wagner, Uli and Mabillard, Isaac and Skopenkov, A. B.}, issn = {0036-0279}, journal = {Russian Mathematical Surveys}, number = {6}, pages = {1156--1158}, publisher = {IOP Publishing}, title = {{Eliminating higher-multiplicity intersections, III. Codimension 2}}, doi = {10.1070/RM9943}, volume = {75}, year = {2020}, } @article{6563, abstract = {This paper presents two algorithms. The first decides the existence of a pointed homotopy between given simplicial maps 𝑓,𝑔:𝑋→𝑌, and the second computes the group [𝛴𝑋,𝑌]∗ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected. More generally, these algorithms work relative to 𝐴⊆𝑋.}, author = {Filakovský, Marek and Vokřínek, Lukas}, issn = {16153383}, journal = {Foundations of Computational Mathematics}, pages = {311--330}, publisher = {Springer Nature}, title = {{Are two given maps homotopic? An algorithmic viewpoint}}, doi = {10.1007/s10208-019-09419-x}, volume = {20}, year = {2020}, } @article{7960, abstract = {Let A={A1,…,An} be a family of sets in the plane. For 0≤i2b be integers. We prove that if each k-wise or (k+1)-wise intersection of sets from A has at most b path-connected components, which all are open, then fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.}, author = {Kalai, Gil and Patakova, Zuzana}, issn = {14320444}, journal = {Discrete and Computational Geometry}, pages = {304--323}, publisher = {Springer Nature}, title = {{Intersection patterns of planar sets}}, doi = {10.1007/s00454-020-00205-z}, volume = {64}, year = {2020}, } @inproceedings{8732, abstract = {A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ , it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles.}, author = {Arroyo Guevara, Alan M and Klute, Fabian and Parada, Irene and Seidel, Raimund and Vogtenhuber, Birgit and Wiedera, Tilo}, booktitle = {Graph-Theoretic Concepts in Computer Science}, isbn = {9783030604394}, issn = {1611-3349}, location = {Leeds, United Kingdom}, pages = {325--338}, publisher = {Springer Nature}, title = {{Inserting one edge into a simple drawing is hard}}, doi = {10.1007/978-3-030-60440-0_26}, volume = {12301}, year = {2020}, } @phdthesis{7944, abstract = {This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.}, author = {Masárová, Zuzana}, isbn = {978-3-99078-005-3}, issn = {2663-337X}, keywords = {reconfiguration, reconfiguration graph, triangulations, flip, constrained triangulations, shellability, piecewise-linear balls, token swapping, trees, coloured weighted token swapping}, pages = {160}, publisher = {Institute of Science and Technology Austria}, title = {{Reconfiguration problems}}, doi = {10.15479/AT:ISTA:7944}, year = {2020}, } @phdthesis{8032, abstract = {Algorithms in computational 3-manifold topology typically take a triangulation as an input and return topological information about the underlying 3-manifold. However, extracting the desired information from a triangulation (e.g., evaluating an invariant) is often computationally very expensive. In recent years this complexity barrier has been successfully tackled in some cases by importing ideas from the theory of parameterized algorithms into the realm of 3-manifolds. Various computationally hard problems were shown to be efficiently solvable for input triangulations that are sufficiently “tree-like.” In this thesis we focus on the key combinatorial parameter in the above context: we consider the treewidth of a compact, orientable 3-manifold, i.e., the smallest treewidth of the dual graph of any triangulation thereof. By building on the work of Scharlemann–Thompson and Scharlemann–Schultens–Saito on generalized Heegaard splittings, and on the work of Jaco–Rubinstein on layered triangulations, we establish quantitative relations between the treewidth and classical topological invariants of a 3-manifold. In particular, among other results, we show that the treewidth of a closed, orientable, irreducible, non-Haken 3-manifold is always within a constant factor of its Heegaard genus.}, author = {Huszár, Kristóf}, isbn = {978-3-99078-006-0}, issn = {2663-337X}, pages = {xviii+120}, publisher = {Institute of Science and Technology Austria}, title = {{Combinatorial width parameters for 3-dimensional manifolds}}, doi = {10.15479/AT:ISTA:8032}, year = {2020}, } @phdthesis{8156, abstract = {We present solutions to several problems originating from geometry and discrete mathematics: existence of equipartitions, maps without Tverberg multiple points, and inscribing quadrilaterals. Equivariant obstruction theory is the natural topological approach to these type of questions. However, for the specific problems we consider it had yielded only partial or no results. We get our results by complementing equivariant obstruction theory with other techniques from topology and geometry.}, author = {Avvakumov, Sergey}, issn = {2663-337X}, pages = {119}, publisher = {Institute of Science and Technology Austria}, title = {{Topological methods in geometry and discrete mathematics}}, doi = {10.15479/AT:ISTA:8156}, year = {2020}, } @inproceedings{15082, abstract = {Two plane drawings of geometric graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common. For a given set S of 2n points two plane drawings of perfect matchings M1 and M2 (which do not need to be disjoint nor compatible) are disjoint tree-compatible if there exists a plane drawing of a spanning tree T on S which is disjoint compatible to both M1 and M2. We show that the graph of all disjoint tree-compatible perfect geometric matchings on 2n points in convex position is connected if and only if 2n ≥ 10. Moreover, in that case the diameter of this graph is either 4 or 5, independent of n.}, author = {Aichholzer, Oswin and Obmann, Julia and Patak, Pavel and Perz, Daniel and Tkadlec, Josef}, booktitle = {36th European Workshop on Computational Geometry}, location = {Würzburg, Germany, Virtual}, title = {{Disjoint tree-compatible plane perfect matchings}}, year = {2020}, } @inproceedings{7401, abstract = {The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest. }, author = {Fulek, Radoslav and Kyncl, Jan}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, isbn = {978-3-95977-104-7}, issn = {1868-8969}, location = {Portland, OR, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Z_2-Genus of graphs and minimum rank of partial symmetric matrices}}, doi = {10.4230/LIPICS.SOCG.2019.39}, volume = {129}, year = {2019}, } @article{5790, abstract = {The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some predrawn chords that represent an induced subgraph of G. The question is whether one can extend R′ to a representation R of the entire graph G, that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of G. Our main result is an O(n3) time algorithm for partial representation extension of circle graphs, where n is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.}, author = {Chaplick, Steven and Fulek, Radoslav and Klavík, Pavel}, issn = {03649024}, journal = {Journal of Graph Theory}, number = {4}, pages = {365--394}, publisher = {Wiley}, title = {{Extending partial representations of circle graphs}}, doi = {10.1002/jgt.22436}, volume = {91}, year = {2019}, } @article{5857, abstract = {A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is [Formula presented](n−1), and that this bound is best possible for infinitely many values of n.}, author = {Fulek, Radoslav and Pach, János}, issn = {0166218X}, journal = {Discrete Applied Mathematics}, number = {4}, pages = {266--231}, publisher = {Elsevier}, title = {{Thrackles: An improved upper bound}}, doi = {10.1016/j.dam.2018.12.025}, volume = {259}, year = {2019}, } @article{6638, abstract = {The crossing number of a graph G is the least number of crossings over all possible drawings of G. We present a structural characterization of graphs with crossing number one.}, author = {Silva, André and Arroyo Guevara, Alan M and Richter, Bruce and Lee, Orlando}, issn = {0012-365X}, journal = {Discrete Mathematics}, number = {11}, pages = {3201--3207}, publisher = {Elsevier}, title = {{Graphs with at most one crossing}}, doi = {10.1016/j.disc.2019.06.031}, volume = {342}, year = {2019}, } @article{7034, abstract = {We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.}, author = {Fulek, Radoslav and Kynčl, Jan}, issn = {1439-6912}, journal = {Combinatorica}, number = {6}, pages = {1267--1279}, publisher = {Springer Nature}, title = {{Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4}}, doi = {10.1007/s00493-019-3905-7}, volume = {39}, year = {2019}, } @article{7108, abstract = {We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.}, author = {Goaoc, Xavier and Patak, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli}, issn = {0004-5411}, journal = {Journal of the ACM}, number = {3}, publisher = {ACM}, title = {{Shellability is NP-complete}}, doi = {10.1145/3314024}, volume = {66}, year = {2019}, } @inproceedings{7230, abstract = {Simple drawings of graphs are those in which each pair of edges share at most one point, either a common endpoint or a proper crossing. In this paper we study the problem of extending a simple drawing D(G) of a graph G by inserting a set of edges from the complement of G into D(G) such that the result is a simple drawing. In the context of rectilinear drawings, the problem is trivial. For pseudolinear drawings, the existence of such an extension follows from Levi’s enlargement lemma. In contrast, we prove that deciding if a given set of edges can be inserted into a simple drawing is NP-complete. Moreover, we show that the maximization version of the problem is APX-hard. We also present a polynomial-time algorithm for deciding whether one edge uv can be inserted into D(G) when {u,v} is a dominating set for the graph G.}, author = {Arroyo Guevara, Alan M and Derka, Martin and Parada, Irene}, booktitle = {27th International Symposium on Graph Drawing and Network Visualization}, isbn = {978-3-0303-5801-3}, issn = {1611-3349}, location = {Prague, Czech Republic}, pages = {230--243}, publisher = {Springer Nature}, title = {{Extending simple drawings}}, doi = {10.1007/978-3-030-35802-0_18}, volume = {11904}, year = {2019}, } @phdthesis{6681, abstract = {The first part of the thesis considers the computational aspects of the homotopy groups πd(X) of a topological space X. It is well known that there is no algorithm to decide whether the fundamental group π1(X) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with π1(X) trivial), compute the higher homotopy group πd(X) for any given d ≥ 2. However, these algorithms come with a caveat: They compute the isomorphism type of πd(X), d ≥ 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of πd(X). We present an algorithm that, given a simply connected space X, computes πd(X) and represents its elements as simplicial maps from suitable triangulations of the d-sphere Sd to X. For fixed d, the algorithm runs in time exponential in size(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected spaces X such that for any simplicial map representing a generator of πd(X), the size of the triangulation of S d on which the map is defined, is exponential in size(X). In the second part of the thesis, we prove that the following question is algorithmically undecidable for d < ⌊3(k+1)/2⌋, k ≥ 5 and (k, d) ̸= (5, 7), which covers essentially everything outside the meta-stable range: Given a finite simplicial complex K of dimension k, decide whether there exists a piecewise-linear (i.e., linear on an arbitrarily fine subdivision of K) embedding f : K ↪→ Rd of K into a d-dimensional Euclidean space.}, author = {Zhechev, Stephan Y}, issn = {2663-337X}, pages = {104}, publisher = {Institute of Science and Technology Austria}, title = {{Algorithmic aspects of homotopy theory and embeddability}}, doi = {10.15479/AT:ISTA:6681}, year = {2019}, } @unpublished{8182, abstract = {Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to {\mathbb R}^n$ whose image avoids the diagonal $\{(x,x\dots,x)\in {\mathbb R}^n|x\in {\mathbb R}\}$. Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $\mathfrak S_n$-equivariant maps from the boundary $\partial\Delta^{n-1}$ of $(n-1)$-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.}, author = {Avvakumov, Sergey and Kudrya, Sergey}, booktitle = {arXiv}, publisher = {arXiv}, title = {{Vanishing of all equivariant obstructions and the mapping degree}}, year = {2019}, }