Topological graph

In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times,no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing). A topological graph is also called a drawing of a graph. In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times,no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing). A topological graph is also called a drawing of a graph. An important special class of topological graphs is the class of geometric graphs, wherethe edges are represented by line segments. (The term geometric graph is sometimes used in a broader, somewhat vague sense.) The theory of topological graphs is an area of graph theory, mainly concerned with combinatorial properties of topological graphs, in particular, with the crossing patterns of their edges. It is closely related to graph drawing, a field which is more application oriented, and topological graph theory, which focuses on embeddings of graphs in surfaces (that is, drawings without crossings). A fundamental problem in extremal graph theory is the following: what is the maximum number of edges that a graph of n vertices can have if it contains no subgraph belonging to a given class of forbidden subgraphs? The prototype of such results is Turán's theorem, where there is one forbidden subgraph: a complete graph with k vertices (k is fixed). Analogous questions can be raised for topological and geometric graphs, with the differencethat now certain geometric subconfigurations are forbidden. Historically, the first instance of such a theorem is due to Paul Erdős, who extendeda result of Heinz Hopf and Erika Pannwitz. He proved that the maximum number of edges that a geometric graph with n > 2 vertices can have without containing two disjoint edges (that cannot even share an endpoint) is n. John Conway conjectured that this statement can be generalized to simple topological graphs. A topological graph is called 'simple' if any pair of its edges share at most one point, which is either an endpoint or a common interior point at which the two edges properly cross. Conway's thrackle conjecture can now be reformulated as follows: A simple topological graph with n > 2 vertices and no two disjoint edges has at most n edges. The first linear upper bound on the number of edges of such a graph was established by Lovász et al..The best known upper bound, 1.428n, was proved by Fulek and Pach. Apart from geometric graphs, Conway's thrackle conjecture is known to be true for x-monotone topological graphs. A topological graph is said to be x-monotone if every vertical line intersects every edge in at most one point. Alon and Erdős initiated the investigation of the generalization of the above question to the case where the forbidden configuration consists of k disjoint edges (k > 2). They proved that the number of edges of a geometric graph of nvertices, containing no 3 disjoint edges is O(n). The optimal bound of roughly 2.5n was determined by Černý.For larger values of k, the first linear upper bound, O ( k 4 n ) {displaystyle O(k^{4}n)} , was established by Pach and Töröcsik. This was improved by Tóth to O ( k 2 n ) {displaystyle O(k^{2}n)} .For the number of edges of a simple topological graphs with no k disjoint edges, only an O ( n log 4 k − 8 ⁡ n ) {displaystyle O(nlog ^{4k-8}n)} upper bound is known. This implies that every complete simple topological graph with n vertices has at least c log ⁡ n log ⁡ log ⁡ n {displaystyle c{frac {log n}{log log n}}} edges, which was improved to c n 1 2 − ϵ {displaystyle cn^{{frac {1}{2}}-epsilon }} by Ruiz-Vargas. It is possible that this lower bound can be further improved to cn, where c > 0 is a constant. A common interior point of two edges, at which the first edge passes from one side of the second edge to the other, is called a crossing. Two edges of a topological graph cross each other if they determine a crossing. For any integer k > 1, a topological or geometric graph is called k-quasi-planar if it has no k pairwise crossing edges.Using this terminology, if a topological graph is 2-quasi-planar, then it is a planar graph.It follows from Euler's polyhedral formula that every planar graph with n > 2 vertices has at most 3n − 6 edges. Therefore, every 2-quasi-planar graph with n > 2 vertices has at most 3n − 6 edges. It has been conjectured by Pach et al. that every k-quasi-planar topological graph with n vertices has at most c(k)nedges, where c(k) is a constant depending only on k. This conjecture is known to be true for k = 3 and k = 4. It is also known to be true for convex geometric graphs (that is for geometric graphswhose vertices form the vertex set of a convex n-gon), and for k-quasi-planar topological graphs whose edges are drawn as x-monotone curves, all of which cross a vertical line .The last result implies that every k-quasi-planar topological graph with n vertices, whose edges are drawn as x-monotone curves has at most c(k)n log n edges for a suitable constant c(k). For geometric graphs, this was proved earlier by Valtr. The best known general upper bound for the number of edges of a k-quasi-planar topological graph is n log O ( log ⁡ k ) ⁡ n {displaystyle nlog ^{O(log k)}n} .