On fan-crossing graphs

Abstract

A fan is a set of edges with a single common endpoint. A drawing of a graph in the plane is fan-crossing if each edge can only be crossed by edges of a fan. It is fan-planar if, in addition, the common endpoint is on the same side of the crossed edge. A drawing is adjacency-crossing if any two edges are adjacent if they cross the same edge. Then multiple independent crossings are excluded, in which an edge is crossed by at least two edges with no common endpoint. In adjacency-crossing drawings it is allowed that an edge crosses the edges of a triangle, which is excluded for fan-crossing drawings. A graph is fan-crossing (fan-planar, adjacency-crossing) if it admits a respective drawing.

We show that every adjacency-crossing graph is fan-crossing and that there are fan-crossing graphs that are not fan-planar. Moreover, for every fan-crossing graph there is a fan-planar graph on the same set of vertices and with the same number of edges. Hence, fan-crossing and fan-planar graphs are different, but they do not differ in the density with at most $5n-10$ edges for graphs of order n.

Introduction

Graphs with or without special patterns for edge crossings are an important topic in Topological Graph Theory, Graph Drawing, and Computational Geometry. Particular patterns are no crossings, single crossings, fans, independent edges, or no three pairwise crossing edges. A fan is a set of edges with a single common endpoint. Edges are independent if they do not share a common endpoint. Important graph classes have been defined by drawings that allow or exclude such crossing patterns, including the planar, 1-planar [14], [16], fan-planar [4], [5], [13], fan-crossing free [10], and quasi-planar graphs [3]. A first order logic definition of these and other graph classes is given in [7], and more general classes are defined by the exclusion of grids [1], [15]. These definitions are motivated by a general interest in particular classes of non-planar graphs [11].

We consider undirected graphs $G=(V,E)$ with finite sets of vertices V and edges E that are simple both in a graph theoretic and in a topological sense. Thus we do not allow multiple edges and self-loops, and we exclude multiple crossings of two edges and crossings among adjacent edges.

A drawing $\mathcal{D}(G)$ of a graph G is a mapping of G into the plane so that the vertices are mapped to distinct points and each edge is mapped to a Jordan curve between the endpoints that does not pass through other endpoints. Two edges cross if their Jordan curves intersect in a point other than an endpoint. Crossings subdivide an edge into uncrossed parts, called segments, whose endpoints are vertices or crossing points. An edge is uncrossed if it consists of a single segment. A drawn graph is called a topological graph. In other works, a topological graph is called an embedding which is the class of topologically equivalent drawings. A drawn graph partitions the plane into topologically connected regions, called faces. The unbounded region is called the outer face. The boundary of each face consists of a cyclic sequence of segments. It is commonly specified by the sequence of vertices and crossing points of the segments. A closed Jordan curve J partitions the plane into two regions, the bounded inner region and the unbounded outer region. Vertices or edges are inside (outside) J if they are drawn in the inner (outer) region. The subgraph of a graph G induced by a subset U of vertices is denoted by $G[U]$. It inherits its drawing from a drawing of G, from which all vertices not in U and all edges with at most one endpoint in U are removed.

Next we consider crossings in drawings of graphs. An edge e has a k-crossing for $k\ge 0$ if it crosses k edges. It is uncrossed if $k=0$. Edge e has a fan-crossing if the set of edges crossing e is a fan, as in Fig. 1(a), and an independent crossing if all crossing edges are independent, which is a multiple independent crossing for $k\ge 2$ edges, see Fig. 1(b). Fan-crossings are also known as biradial grid crossings [15] and independent crossings as natural grid crossings [1]. An edge has an adjacency-crossing if all crossing edges are pairwise adjacent. Thus multiple independent crossing and adjacency-crossing are complementary [7], whereas fan-crossing and independent crossing drawings both admit edges that are crossed at most once.

Fan-planar drawings were introduced by Kaufmann and Ueckerdt [13], who imposed a special restriction on fan-crossing drawings, called configuration II. It is shown in Fig. 2(a). Let $e,f$ and g be three edges in a drawing so that e is crossed by f and g, and f and g share a common vertex t. Then they form configuration II if one endpoint of e is inside a closed Jordan curve through t with parts of $e,f$ and g, and the other endpoint of e is outside this curve. If $e=\{u,v\}$ is oriented from u (left) to v (right) and f and g are oriented away from t, then f and g cross e from different directions and vertex t is not on the same side of e. There is configuration II if there is a triangle-crossing in which an edge crosses the edges of a triangle, see Fig. 2(b). Note that a triangle-crossing is the only configuration, in which an edge is crossed by edges that do not form a fan and no two of which are independent.

Observe the subtle differences between adjacency-crossing, fan-crossing, and fan-planar drawings, which each exclude multiple independent crossings. In addition, fan-crossing drawings exclude triangle-crossings, and fan-planar drawings exclude configuration II. Kaufmann and Ueckerdt [13] observed that configuration II cannot occur in straight-line drawings. Hence, every straight-line adjacency-crossing drawing is fan-planar.

It is common to define graphs by properties of their drawings. A graph is fan-planar if it admits a fan-planar drawing. Fan-crossing, adjacency-crossing, k-planar and independent crossing graphs are defined accordingly, where independent crossing graphs are called fan-crossing free [10]. A fan-crossing free graph admits a drawing so that the set of edges crossed by an edge does not contain a fan with at least two edges. Clearly, every 1-planar graph is both fan-crossing and fan-crossing free, but not conversely as shown in [8].

Kaufmann and Ueckerdt [13] proved that fan-planar graphs of order $n\ge 3$ have at most $5n-10$ edges and posed the density of adjacency-crossing graphs as an open problem. The density defines an upper bound on the number of edges in graphs of order n.

We answer the above question and prove the following:

• Every adjacency-crossing graph is fan-crossing. Thus triangle-crossings can be avoided in adjacency-crossing drawings.

• There are fan-crossings graphs that are not fan-planar. Thus configuration II is essential and cannot be avoided in some fan-crossing drawings.

• For every fan-crossing graph G there is a fan-planar graph $G^{\prime }$ on the same set of vertices and with the same number of edges. Thus fan-crossing graphs of order $n\ge 3$ have at most $5n-10$ edges.

Our tool is edge rerouting. A rerouted edge is denoted by $\tilde{e}$ if e is the original one. A rerouted edge $\tilde{e}=\{u,v\}$ first follows an edge incident to u to a crossing with an edge f, then f and probably a sequence of edges, where an edge and its successor cross, and finally an edge incident to v, as illustrated in Fig. 3. Here e may be used first or last. More formally, by rerouting one or more edges, we transform an adjacency-crossing (fan-crossing) drawing $\mathcal{D}(G)$ into an adjacency-crossing (fan-crossing) drawing $\tilde{\mathcal{D}}(G)$, which differs from $\mathcal{D}(G)$ in the drawing of the rerouted edges. The rerouting is good if it does not create multiple independent crossings, at least one edge is no longer crossed from opposite directions after its rerouting, and any rerouted edge does not create new edge crossings from opposite directions. In other words, a good edge rerouting is a step towards a fan-planar drawing.

In the rest of the paper, we prove that triangle-crossings can be circumvented by an edge rerouting in Section 2 and study configuration II in Section 3. We conclude in Section 4 with some open problems on fan-crossing graphs.