TY - CONF
AB - matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.
AU - Aichholzer, Oswin
AU - Arroyo Guevara, Alan M
AU - Masárová, Zuzana
AU - Parada, Irene
AU - Perz, Daniel
AU - Pilz, Alexander
AU - Tkadlec, Josef
AU - Vogtenhuber, Birgit
ID - 9296
SN - 03029743
T2 - 15th International Conference on Algorithms and Computation
TI - On compatible matchings
VL - 12635
ER -
TY - JOUR
AB - Hill's Conjecture states that the crossing number cr(𝐾𝑛) of the complete graph 𝐾𝑛 in the plane (equivalently, the sphere) is 14⌊𝑛2⌋⌊𝑛−12⌋⌊𝑛−22⌋⌊𝑛−32⌋=𝑛4/64+𝑂(𝑛3) . Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely 𝑛4/64+𝑂(𝑛3) , thus matching asymptotically the conjectured value of cr(𝐾𝑛) . Let cr𝑃(𝐺) denote the crossing number of a graph 𝐺 in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of 𝐾𝑛 is (𝑛4/8𝜋2)+𝑂(𝑛3) . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if lim𝑛→∞ cr𝑃(𝐾𝑛)/𝑛4=1/8𝜋2 . We construct drawings of 𝐾𝑛 in the projective plane that disprove this.
AU - Arroyo Guevara, Alan M
AU - Mcquillan, Dan
AU - Richter, R. Bruce
AU - Salazar, Gelasio
AU - Sullivan, Matthew
ID - 9295
JF - Journal of Graph Theory
SN - 0364-9024
TI - Drawings of complete graphs in the projective plane
ER -
TY - CONF
AB - 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.
AU - Arroyo Guevara, Alan M
AU - Bensmail, Julien
AU - Bruce Richter, R.
ID - 7994
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Extending drawings of graphs to arrangements of pseudolines
VL - 164
ER -
TY - CONF
AB - 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.
AU - Arroyo Guevara, Alan M
AU - Klute, Fabian
AU - Parada, Irene
AU - Seidel, Raimund
AU - Vogtenhuber, Birgit
AU - Wiedera, Tilo
ID - 8732
SN - 0302-9743
T2 - Graph-Theoretic Concepts in Computer Science
TI - Inserting one edge into a simple drawing is hard
VL - 12301
ER -
TY - CONF
AB - 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.
AU - Arroyo Guevara, Alan M
AU - Derka, Martin
AU - Parada, Irene
ID - 7230
SN - 03029743
T2 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
TI - Extending simple drawings
VL - 11904
ER -
TY - JOUR
AB - 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.
AU - Silva, André
AU - Arroyo Guevara, Alan M
AU - Richter, Bruce
AU - Lee, Orlando
ID - 6638
IS - 11
JF - Discrete Mathematics
SN - 0012-365X
TI - Graphs with at most one crossing
VL - 342
ER -