TY - JOUR
AB - Hill's Conjecture states that the crossing number cr(πΎπ) of the complete graph πΎπ in the plane (equivalently, the sphere) is 14βπ2ββπβ12ββπβ22ββπβ32β=π4/64+π(π3) . 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 π4/64+π(π3) , thus matching asymptotically the conjectured value of cr(πΎπ) . Let crπ(πΊ) denote the crossing number of a graph πΊ in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of πΎπ is (π4/8π2)+π(π3) . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if limπβββcrπ(πΎπ)/π4=1/8π2 . We construct drawings of πΎπ 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 - JOUR
AB - Motivated by the successful application of geometry to proving the Harary--Hill conjecture for βpseudolinearβ drawings of $K_n$, we introduce βpseudosphericalβ drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points---no three on a great circle---and the edges of $G$ are shortest-arcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $\gamma_e$ such that the only vertices in $\gamma_e$ are the ends of $e$; (2) if $e\ne f$, then $\gamma_e\cap\gamma_f$ has precisely two crossings; and (3) if $e\ne f$, then $e$ intersects $\gamma_f$ at most once, in either a crossing or an end of $e$. We use properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that for the complete graph, properties (1)--(3) are equivalent to the same three properties but with βprecisely two crossingsβ in (2) replaced by βat most two crossings.β The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs (coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.
AU - Arroyo Guevara, Alan M
AU - Richter, R. Bruce
AU - Sunohara, Matthew
ID - 9468
IS - 2
JF - SIAM Journal on Discrete Mathematics
SN - 08954801
TI - Extending drawings of complete graphs into arrangements of pseudocircles
VL - 35
ER -
TY - CONF
AB - matching is compatible to two or more labeled point sets of size n with labels {1,β¦,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 β2nββββ edges. More generally, for any β labeled point sets we construct compatible matchings of size Ξ©(n1/β) . As a corresponding upper bound, we use probabilistic arguments to show that for any β given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(β+1)) edges. Finally, we show that Ξ(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 - MasΓ‘rovΓ‘, 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 - 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 Leviβs 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 Ο , it can be decided in polynomial time whether there exists a pseudocircle Ξ¦Ο extending Ο for which Aβͺ{Ξ¦Ο} 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 Leviβs 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, AndrΓ©
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 -