Crossings In Drawing Research Articles

On the Maximum Number of Crossings in Star-Simple Drawings of $K_n$ with No Empty Lens

A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. We forbid empty lenses, i.e., every lens is required to enclose a vertex, and show that with this restriction $3\cdot(n-4)!$ is an upper bound on the number of crossings between two edges of a star-simple drawing of $K_n$. It follows that $n!$ bounds the total number of crossings in the drawing. This is the first finite upper bound on the number of crossings in star-simple drawings of the complete graph $K_n$ with no empty lens. For a lower bound we construct a star-simple drawing of $K_n$ with no empty lens in which a pair of edges contributes $5^{n/2-2}$ crossings.

The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

A bipartite-cylindrical drawing of the complete bipartite graph $$ K_{m,n} $$ is a drawing on the surface of a cylinder, where the vertices are placed on the boundaries of the cylinder, one vertex-partition per boundary, and the edges do not cross the boundaries. The bipartite-cylindrical crossing number of $$ K_{m,n}$$, denoted by $$ cr_\circledcirc (K_{m,n}) $$, is the minimum number of crossings among all bipartite-cylindrical drawings of $$ K_{m,n} $$. This problem is equivalent to minimizing the number of crossings in drawings of $$ K_{m,n} $$ in the plane where each of the vertex classes in the bipartition is placed on a circle and the edges do not cross the two circles. We determine $$ cr_\circledcirc (K_{m,n}) $$ and give explicit constructions that achieve this number. This in turn gives an alternative proof to the Harary–Hill Conjecture restricted to a subclass of cylindrical drawings of the complete graph.

Räume und Grenzen in der Laienmetasprache. Eine Metaphernanalyse zu Sprache und Sprecher

In folk linguistic studies people provide information about their perception and evaluation of languages and their speakers. In so doing, they use different strategies of verbalization. Frequently, they conceptualize space, boundaries, and crossings in drawing on mental concepts as metaphors and metonymies. This paper shows, based on a study on language attitudes in German-speaking Switzerland, the strategies laypeople use to organize their mental (language) spaces. For this purpose, I shall discuss recent data, collected via questionnaires and interviews, which illustrate how language attitudes are conceptualized and verbalized through metaphors and metonymies, and how the relationship between language and speaker is established through the process of indexicalization and iconization.

On Crossing Numbers of Complete Tripartite and Balanced Complete Multipartite Graphs

AbstractThe crossing number cr(G) of a graph G is the minimum number of crossings in a drawing of G in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number of G is the minimum number of crossings in a such drawing of G with edges as straight line segments. Zarankiewicz proved in 1952 that . We generalize the upper bound to urn:x-wiley:03649024:media:jgt22041:jgt22041-math-0004and prove . We also show that for n large enough, and , with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r‐partite graph. Richter and Thomassen proved in 1997 that the limit as of over the maximum number of crossings in a drawing of exists and is at most . We define and show that for a fixed r and the balanced complete r‐partite graph, is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.

The 2-Page Crossing Number of $$K_{n}$$ K n

Around 1958, Hill described how to draw the complete graph $$K_n$$ with $$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$ crossings, and conjectured that the crossing number $${{\mathrm{cr}}}(K_{n})$$ of $$K_n$$ is exactly $$Z(n)$$ . This is also known as Guy’s conjecture as he later popularized it. Towards the end of the century, substantially different drawings of $$K_{n}$$ with $$Z(n)$$ crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line $$\ell $$ (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by $$\ell $$ . The 2-page crossing number of $$K_{n} $$ , denoted by $$\nu _{2}(K_{n})$$ , is the minimum number of crossings determined by a 2-page book drawing of $$K_{n}$$ . Since $${{\mathrm{cr}}}(K_{n}) \le \nu _{2}(K_{n})$$ and $$\nu _{2}(K_{n}) \le Z(n)$$ , a natural step towards Hill’s Conjecture is the weaker conjecture $$\nu _{2}(K_{n}) = Z(n)$$ , popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of $$K_{n}$$ , and use it to prove that $$\nu _{2}(K_{n}) = Z(n) $$ . To this end, we extend the inherent geometric definition of $$k$$ -edges for finite sets of points in the plane to topological drawings of $$K_{n}$$ . We also introduce the concept of $${\le }{\le }k$$ -edges as a useful generalization of $${\le }k$$ -edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of $$K_{n}$$ in terms of its number of $${\le }k$$ -edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of $$K_{n}$$ and show that, up to equivalence, they are unique for $$n$$ even, but that there exist an exponential number of non homeomorphic such drawings for $$n$$ odd.

Pfaffian graphs, T-joins and crossing numbers

We characterize Pfaffian graphs in terms of their drawings in the plane. We generalize the techniques used in the proof of this characterization, and prove a theorem about the numbers of crossings in T-joins in different drawings of a fixed graph. As a corollary we give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K 2j+1 and K 2j+1,2k+1.

Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas

We give improved approximations for two classical embedding problems: (i) minimizing the number of crossings in a drawing on the plane of a bounded degree graph; and (ii) minimizing the VLSI layout area of a graph of maximum degree four. These improved algorithms can be applied to improve a variety of VLSI layout problems. Our results are as follows. (i) We compute a drawing on the plane of a bounded degree graph in which the sum of the numbers of vertices and crossings is O(log3 n)$ times the optimal minimum sum. This is a logarithmic factor improvement relative to the best known result. (ii) We compute a VLSI layout of a graph of maximum degree four in a square grid whose area is O(log4 n)$ times the minimum layout area. This is an O(log2 n) improvement over the best known long-standing result.

The monadic second-order logic of graphs XIII: Graph drawings with edge crossings

We introduce finite relational structures called sketches, that represent edge crossings in drawings of finite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges that may cross one another and that may cross the edges of the frame. We prove the 3- Edge Theorem stating that a structure of appropriate type with a frame is a sketch if and only if every induced substructure representing the frame and at most 3 edges not in the frame is a sketch.

A neural-network algorithm for a graph layout problem

We present a neural-network algorithm for minimizing edge crossings in drawings of nonplanar graphs. This is an important subproblem encountered in graph layout. The algorithm finds either the minimum number of crossings or an approximation thereof and also provides a linear embedding realizing the number of crossings found. The parallel time complexity of the algorithm is O(1) for a neural network with n(2) processing elements, where n is the number of vertices of the graph. We present results from testing a sequential simulator of the algorithm on a set of nonplanar graphs and compare its performance with the heuristic of Nicholson.

Edges without crossings in drawings of complete graphs

In drawings (two edges have at most one point in common, either a node or a crossing) of the complete graph K n in the Euclidean plane there occur at most 2 n − 2 edges without crossings. This was proved by G. Ringel in [1]. Here the minimal number of edges without crossings in drawings of K n is determined, and for the existence of values between minimum and maximum is asked.

The crossing number of K 5,n

Several arguments are presented which provide restrictions on the possible number of crossings in drawings of bipartite graphs. In particular it is shown that cr(K 5,n) =4[1/2 n][1/2( n−1)] and cr(K 6,n) =6[1/2 n][1/2( n−1)].