Simple Topological Graph Research Articles

Graph Planarity by Replacing Cliques with Paths

This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.

The biplanar tree graph

The complete twisted graph of order n, denoted by $$T_n$$ , is a complete simple topological graph with vertices $$u_1, u_2, \ldots , u_n$$ such that two edges $$u_iu_j$$ and $$u_{i'}u_{j'}$$ cross if and only if $$ i<i'<j'<j $$ or $$ i'<i<j<j' $$ . The convex geometric complete graph of order n, denoted by $$G_n$$ , is a convex geometric graph with vertices $$v_1, v_2, \ldots , v_n$$ placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set $$\{v_1,v_2,\ldots , v_n \}$$ having the property of being planar when embedded in both $$T_n$$ and $$G_n$$ . Given a connected graph G the (combinatorial) tree graph $$\mathcal {T}(G)$$ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in $$\mathcal {T}(G)$$ if there are edges $$e \in P$$ and $$f\in Q$$ such that $$Q=P - e + f$$ . For all positive integers n, we denote by $$\mathcal {T}(n)$$ the graph $$\mathcal {T}(K_n)$$ . The biplanar tree graph, $$\mathcal {B}(n)$$ , is the subgraph of $$\mathcal {T}(n)$$ induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.

Polyline drawings with topological constraints

We study the problem of representing topological graphs as polyline drawings with few bends per edge and such that the topology of the graph is either fully or partially preserved. More formally, let G be a simple topological graph and let Γ be a polyline drawing of G. Drawing Γ partially preserves the topology of G if it has the same external boundary, the same circular order of the edges around each vertex, and the same set of crossings as G, while it fully preserves the topology of G if the planarization of G and the planarization of Γ have the same planar embedding. We prove that if the set of crossing-free edges of G forms a biconnected (connected) spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most one (three), i.e., with at most one (three) bend(s) per edge. If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Ω(n), while it is O(1) if the number of connected components is O(1). Concerning drawings that fully preserve the topology, we show that if G is k-skew (i.e., it becomes planar after removing k suitably chosen edges), it admits one such drawing with curve complexity at most 2k; for 1-skew graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs (i.e., with at most two crossings per edge and maximum edge density), for which we discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

Simple k-planar graphs are simple (k + 1)-quasiplanar

A simple topological graph is $k$-quasiplanar ($k\geq 2$) if it contains no $k$ pairwise crossing edges, and $k$-planar if no edge is crossed more than $k$ times. In this paper, we explore the relationship between $k$-planarity and $k$-quasiplanarity to show that, for $k \geq 2$, every $k$-planar simple topological graph can be transformed into a $(k+1)$-quasiplanar simple topological graph.

Saturated simple and 2-simple topological graphs with few edges

A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a $k$-simple topological graph, every pair of edges has at most $k$ common points of this kind. We construct saturated simple and 2-simple graphs with few edges. These are $k$-simple graphs in which no further edge can be added. We improve the previous upper bounds of Kynčl, Pach, Radoičić, and Tóth [Comput. Geom., 48, 2015] and show that there are saturated simple graphs on $n$ vertices with only $7n$ edges and saturated 2-simple graphs on $n$ vertices with $14.5n$ edges. As a consequence, there is a $k$-simple graph (for a general $k$), which can be saturated using $14.5n$ edges, while previous upper bounds suggested $17.5n$ edges. We also construct saturated simple and 2-simple graphs that have some vertices with low degree.

Many disjoint edges in topological graphs

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on n vertices contains Ω(n1−ϵ) pairwise disjoint edges for any ϵ>0. As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains Ω(n12−ϵ) pairwise disjoint edges for any ϵ>0. This improves the previous lower bound of Ω(n13) by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges.

Many disjoint edges in topological graphs

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on n vertices contains Ω(n1−ϵ) pairwise disjoint edges for any ϵ>0. As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains Ω(n12−ϵ) pairwise disjoint edges for any ϵ>0. This improves the previous lower bound of Ω(n13) by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges.

Disjoint edges in topological graphs and the tangled-thrackle conjecture

It is shown that for a constant t∈N, every simple topological graph on n vertices has O(n) edges if the graph has no two sets of t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is Kt,t-free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoičić, and Tóth: Every n-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most O(n) edges.

Empty Triangles in Complete Topological Graphs

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Let $$G$$G be a complete simple topological graph on $$n$$n vertices. The three edges induced by any triplet of vertices in $$G$$G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that $$G$$G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least $$2n/3$$2n/3. We settle Harborth's conjecture in the affirmative.

Saturated simple and k-simple topological graphs

A simple topological graph G is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. G is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with n vertices and O(n) edges. For every k>1, we give similar constructions for k-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most k points in common. We show that in any k-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most 2k times. Another construction shows that the bound 2k cannot be improved. Several other related problems are also considered.

Estimating the Number of Disjoint Edges in Simple Topological Graphs via Cylindrical Drawings

A topological graph drawn on a cylinder whose base is horizontal is angularly monotone if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone drawing of a complete graph on $n$ vertices contains at least $c$ pairwise disjoint edges. We show that for every simple complete topological graph $G$ there exists $\Delta$, $0<\Delta<n$, such that $G$ contains at least $\max \{\frac n\Delta, c(\Delta)\}$ pairwise disjoint edges. By combining our result with a result of Toth we obtain an alternative proof for the best known lower bound of $\Omega(n^\frac 13)$ on the maximum number of pairwise disjoint edges in a simple complete topological graph proved by Suk. Our proof is based on a result of Ruiz-Vargas.

Empty triangles in complete topological graphs

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Let G be a simple complete topological graph. The three edges induced by any triple of vertices in G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth [Harborth, Heiko, Empty triangles in drawings of the complete graph, Discrete Mathematics, 191 (1998), 109–111; Brass, Peter, William O. J. Moser and János Pach, “Research Problems in Discrete Geometry” Springer, (2005)] proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. In this note, we verify Harborthʼs conjecture.

Improved Enumeration of Simple Topological Graphs

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^O(n^2 log(m/n)), and at most 2^O(mn^{1/2} log n) if m < n^{3/2}. As a consequence we obtain a new upper bound 2^O(n^{3/2} log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^O(1)}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^Omega(m^2) for graphs with m > (6+epsilon)n.

On the Degenerate Crossing Number

The degenerate crossing number\({\text{ cr}^{*}}(G)\) of a graph \(G\) is the minimum number of crossing points of edges in any drawing of \(G\) as a simple topological graph in the plane. This notion was introduced by Pach and Toth who showed that for a graph \(G\) with \(n\) vertices and \(e \ge 4n\) edges \({\text{ cr}^{*}}(G)=\Omega \big (e^4 / n^4\big )\). In this paper we completely resolve the main open question about degenerate crossing numbers and show that \({\text{ cr}^{*}}(G)=\Omega \big (e^3 / n^2 \big )\), provided that \(e \ge 4n\). This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.

Disjoint Edges in Complete Topological Graphs

It is shown that every complete $$n$$ -vertex simple topological graph has at $$\varOmega (n^{1/3})$$ pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and Toth, which appears as Problem 5 from Chapter 9.5 in Research Problems in Discrete Geometry by Brass, Moser, and Pach.

Disjoint edges in topological graphs

A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.

Enumeration of simple complete topological graphs

A simple topological graph T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is 2 Θ ( n 4 ) . We also show that the number of weak isomorphism classes of simple complete topological graphs with n vertices and n 4 crossings is at least 2 n ( log n − O ( 1 ) ) , which improves the estimate of Harborth and Mengersen.

On edges crossing few other edges in simple topological complete graphs

We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graph T = ( V , E ) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple if any two edges meet in at most one common point. Let h = h ( n ) be the smallest integer such that every simple topological complete graph on n vertices contains an edge crossing at most h other edges. We show that Ω ( n 3 / 2 ) ≤ h ( n ) ≤ O ( n 2 / log 1 / 4 n ) . We also show that the analogous function on other surfaces (torus, Klein bottle) grows as c n 2 .

Enumeration of simple complete topological graphs

Abstract A simple topological graph T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is 2 Θ ( n 4 ) . We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and ( n 4 ) crossings is at least 2 n ( log n − O ( 1 ) ) , which improves the estimate of Harborth an Mengersen.

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1–9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8 n edges for n vertices. Our best construction with 7 n − O ( 1 ) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5 n − O ( 1 ) , thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.