Journal of Graph Algorithms and Applications 1-visibility Representations of 1-planar Graphs

A 1-visibility representation of a graph displays each vertex as a horizontal vertex-segment, called a bar, and each edge as a vertical edge-segment between the segments of the vertices, such that each edge-segment crosses at most one vertex-segment and each vertex-segment is crossed by at most one edge-segment. A graph is 1-visible if it has such a representation. 1-visibility is related to 1-planarity where graphs are drawn such that each edge is crossed at most once, and specializes bar 1-visibility where vertex-segments can be crossed many times. We develop a linear time algorithm to compute a 1-visibility representation of an embedded 1-planar graph in O(n 2) area. Hence, every 1-planar graph is 1-visible. Concerning density, both 1-visible and 1-planar graphs of size n have at most 4n − 8 edges. However, for every n ≥ 7 there are 1-visible graphs with 4n − 8 edges, which are not 1-planar.


Introduction
Drawing planar graphs is an important topic in graph theory, combinatorics, and in particular in graph drawing. The existence of straight-line drawings was independently proved by Wagner [42], Steinitz and Rademacher [36], Stein [35] and Fáry [21]. The stunning results of de Fraysseix, Pach and Pollack [10] and Schnyder [34] show that planar graphs admit straight-line grid drawings in quadratic area, which can be computed in linear time.
A visibility representation is another way to draw a planar graph. Here the vertices are drawn as horizontal bars, called vertex-segments, and two vertexsegments must see each other along a vertical line, called edge-segment, if there is an edge between the respective vertices. Vertex-and edge-segments do not overlap except that an edge-segment begins and ends at the vertex-segments of its end vertices. For convenience, we identify vertices and edges with their segments. Otten and van Wyck [29] showed that every planar graph has a visibility representation, and a linear time algorithm for constructing it was given independently by Rosenstiehl and Tarjan [33] and by Tamassia and Tollis [40]. Their algorithm uses a grid of size at most (2n − 5) × (n − 1), which was gradually improved to ( 4n/3 − 2) × (n − 1) [20].
Visibility representations lead to clear pictures and have gained a lot of interest, see also [14] and the references given there. For planar graphs there are three (main) versions of visibility: weak, , and strong. They differ in the representation of the segments. In the weak version the vertex-segments may or may not include their extremes, such that an edge-segment may pass the open end of a third unrelated one. Two vertices must see each other if they are adjacent, but not conversely. Hence, weak visibility preserves the subgraph property, which says that every subgraph of a weak visibility graph is a weak visibility graph. In the -version, the edge-segments are bands with thickness > 0 and there is an edge if and only if the corresponding vertices see each other. Finally, in the strong version there is an edge if and only if there is a visibility. The latter makes an essential difference, since the K 2,3 has no strong visibility representation. Moreover, it is N P-hard to determine whether a 3-connected planar graph has a strong visibility representation [2], whereas weak visibility is equivalent to planarity and thus testable in linear time. Weak and -visibility coincide on 2-connected planar graphs, and every 4-connected planar graph has a strong visibility representation [40].
There are several attempts to generalize the planar graphs to nearly planar graphs, e.g., via forbidden minors [32], surfaces of higher genus, or various restrictions on crossings, such as k-planar [30], almost planar [23] or right angle crossing (RAC) graphs [15]. Here, we consider 1-planar graphs, which are defined by drawings in the plane such that each edge is crossed at most once. 1-planar graphs were introduced by Ringel [31] and occur when a planar graph and its dual are drawn simultaneously [18]. As an example consider the complete graph K 6 , which can be drawn 1-planar with two nested triangles and straight-line edges.
The straight-line or rectilinear drawability of 1-planar graphs was first investigated by Eggleton [17]. He settled this problem for outer 1-planar graphs and proved that every outer 1-planar graph has a straight-line drawing. In outer 1-planar graphs all vertices are in the outer face and each edge is crossed at most once. Thomassen [41] generalized this result and proved that an embedded 1-planar graph has a straight-line drawing if and only if it excludes B-and Wconfigurations, see Figs. 1(a) and 1(b). Then only X-configurations remain for pairs of crossing edges, see Fig. 1(c). The forbidden configurations were rediscovered by Hong et al. [25], who also showed that there is a linear time algorithm to convert a 1-planar embedding without these forbidden configurations into a straight-line drawing. In fact, 1-planar graphs can be drawn as nice as planar graphs. Alam et al. [1] proved that every 3-connected 1-planar graph has an embedding with at most one W-configuration in the outer face, and has a straight-line grid drawing in quadratic area with the exception of a single edge in the outer face. Such drawings can be computed in linear time from a given 1-planar embedding as a witness for 1-planarity. Here we add visibility representations. There is a close relationship between 1-planar graphs and right angle crossing (RAC) graphs, where edges must be straight-line and cross at a right angle [15]. 1-planar graphs and RAC graphs have almost the same density, i.e., the maximal number of edges for graphs of size n, namely 4n − 8 and 4n − 10. Eades and Liotta [16] proved that every maximally dense RAC graph is 1-planar. Conversely, every outer 1-planar graph has a RAC drawing with the same embedding [13]. Hence, the RAC graphs range between the outer 1-planar and the 1-planar graphs. In fact, outer 1-planar graphs are planar [3].
Visibility representations have variously been generalized to two dimensions with vertices as non-overlapping paraxial rectangles and edges represented by horizontal and vertical visibility. In the rectangle visibility approach [12,26,27] horizontal and vertical edge-segments may cross and the resulting graphs have up to 6n−20 edges. Horizontal and vertical lines for edges were allowed in Biedl's flat visibility representation [5], however, the lines do not cross and the horizontal lines are a shortcut for a local adjacency. Hence, the concept is equivalent to weak visibility of planar graphs. The term 1-and 2-visibility was used by Fößmeier et al. [22] for orthogonal drawings of planar graphs.
Dean et al. [11] introduced k-bar visibility, where the vertices are represented as horizontal bars and bars are allowed to see through at most k other bars. Thus 0-bar visibility is the common planar visibility, and in 1-visibility a bar can be crossed by the visibility lines of many other bars. They discussed the weak, and strong versions and showed that 1-bar visible graphs have at most 6n − 20 edges. In fact, the formula for the density indicates that k-bar visible graphs are related to k-quasi-planar graphs [23,37], where no k + 2 edges cross mutually. Recently, Sultana et al. [38] showed that some special classes of graphs including the maximal outer 1-planar graphs are 1-bar visible.
In this paper we generalize visibility representations such that they capture 1-planarity. The vertices are drawn as horizontal vertex-segments and an edge needs a vertical visibility and is represented by an edge-segment. Uncrossed segments are transparent and become impermeable if they are crossed by a segment of the other type. Hence, all crossings are right angle crossings (RAC) between a vertex and an edge, and each object is involved in at most one crossing.
We show that every 1-planar graph has a 1-visibility representation in O(n 2 ) area, which can be computed in linear time from a given 1-planar embedding as a witness for 1-planarity. This settles a conjecture of Sultana et al. [38]. The algorithm uses the standard technique for visibility representations of planar graphs from [14,33,40] via the st-numbering of the graph and its dual, which operates on the planar skeleton without crossing edges. The given embedding is augmented and transformed such that the 3-connected components have a normalized embedding [1] and are separated by a copy of the edge between the separation pair. A local transformation suffices to re-insert a pair of crossing edges into the face left by their extraction. The 3-connected components are sandwiched between the horizontal vertex-segments of their separation pair, which comes directly from the st-numberings.
1-visible graphs have the same maximal density as 1-planar graphs with at most 4n − 8 edges for graphs of size n. This is readily seen, since a 1-visible graphs consists of a planar subgraph together with one crossing edge per vertex. Since the two outermost vertices are excluded the density reaches at most 4n − 8. So we provide a new and simple proof of the maximal density of 1-planar graphs. The so-called extended wheel graphs XQ k [7] are examples of 1-planar graphs with maximal density. The XQ 8 graph is shown in Fig. 2, where the visibility representation is obtained by our algorithm. However, there are 1- visibility graphs with 4n−8 edges which are not 1-planar, including the complete graph on 7 vertices without one edge, K 7 -e, which is not 1-planar [7,39]. Hence, the 1-visible graphs properly include the 1-planar graphs, even for maximally dense graphs.

Preliminaries
Consider simple undirected graphs G = (V, E) with n vertices and m edges. We suppose that the graphs are 2-connected, otherwise, the components are treated separately, and are placed next to each other as in [40]. Note that articulation points may cause problems in visibility representations and make the difference between weak and -visibility. This difference vanishes if the articulation points are in one face [40]. Articulation points do not matter in the weak version of visibility.
A drawing of a graph is a mapping of G into the plane such that the vertices are mapped to distinct points and each edge is a Jordan arc between its endpoints. A drawing is planar if the Jordan arcs of the edges do not cross and it is 1-planar if each edge is crossed at most once. In 1-planar drawings crossings of edges with the same endpoint are excluded.
An embedding E(G) of a planar graph G specifies faces. A face is a topologically connected region and is given by a cyclic sequence of edges and vertices that forms its boundary. One of the faces is unbounded and is called the outer face.
Accordingly, a 1-planar embedding E(G) specifies the faces in a 1-planar drawing of a graph G including the outer face. A 1-planar embedding is a witness for 1-planarity. In particular, it describes the pairs of crossing edges and the face where the edges cross. Here a face is given by a cyclic list of edges and half-edges and their vertices and crossing points. A half-edge is a segment of an edge from a vertex to a crossing point. Each crossing point in a 1-planar embedding is incident to four half-edges. If the crossing points are taken as new vertices and the halfedges as edges, then we have the planarization of E(G), which is an embedded planar graph. This structure is used by algorithms operating on E(G), where crossing points always remain as vertices of degree four and may need a special treatment. Two 1-planar (planar) embeddings E 1 (G) and E 2 (G) of a graph are equivalent if there is a homeomorphism h on the plane with E 2 (G) = h(E 1 (G)). Then one embedding can be transformed into the other while preserving all faces including the outer face. Such transformations are embedding preserving.
A visibility representation of a planar graph displays the vertices as horizontal bars, called vertex-segments, and two bars must see each other along a vertical edge-segment if there is an edge between the respective vertices. This is the weak version of visibility, where vertex-segments can see each other but their vertices are not necessarily connected by an edge.
In this paper, we generalize visibility representations such that they fit to 1-planar graphs. We use weak visibility, since we wish to preserve the subgraph property: every subgraph of a 1-visible graph is 1-visible. The vertex-segments include their extremes and start and end at grid points. The and strong versions of 1-visibility do not seem useful, since many planar graphs cannot be represented that way, such as circles of length at least four. The if and only if condition between edges and 1-visibility enforces at least one chord between non-adjacent vertices.
Definition 1. A 1-visibility representation of a graph G = (V, E) displays each vertex v as a horizontal vertex-segment Γ (v) and each edge e = (u, v) as a vertical edge-segment Γ (e) from some point on Γ (u) to some point on Γ (v). The endpoints of all segments are grid points. Vertex segments (edge segments) do not overlap (in their interior). Each vertex-segment is crossed by at most one edge-segment and each edge-segment may cross at most one vertex-segment.
Notice that 1-visibility drawings are straight-line drawings on grids and there are right angle crossings between edges and vertices. Hence, we have a new type of RAC drawings [15,16].

Basic Properties
A 1-planar embedding is planar maximal if no further edge can be added without inducing a crossing or multiple edges. A 1-planar embedding can be augmented to a planar maximal embedding via its planarization, where crossing points remain as vertices of degree four. The augmentation can be computed in linear time from the embedding. Note that the maximality depends on the embedding and a different embedding of a graph may give rise to another maximal planar augmentation, as the transformation of a B-configuration in Fig. 1(a) into a X-configuration in Fig. 1(c) illustrates. In X-configurations all four vertices may have outer neighbors and there are at most three such vertices in a B-configuration. Planar maximal embeddings have nice properties. Proof. The first statement is due to the fact that missing edges between the end vertices can be routed near to the crossing edges. This has been stated at several places, first of all in [7]. For (2) the chords of a pentagon cannot be realized in a single inner or outer face such that each chord has at most one crossing, whereas a quadrangle can be realized as shown by B-configuration with an inner face with four vertices, see Fig. 1(a). Accordingly, there cannot be more than 8 half-edges in a face of a planar maximal 1-planar embedding, see Fig. 3, which also implies (4). Similar properties were established for 1-planar embeddings without B-and W-configurations by Hong et al. [25] and by Alam et al. [1] for 3-connected 1planar graphs. Moreover, the faces can be simplified if the embedding is changed. Then edges cross internally as in an augmented X-configuration or externally as in a W-configurations, and there are at most two crossing points per face.
Eggleton [17] raised the problem which 1-planar graphs have drawings with straight-line edges. He solved this problem for outerplanar graphs, where all crossing points are internal and appear in X-configurations. Thomassen [41] characterized the rectilinear 1-planar embeddings by the exclusion of B-and W-configurations, which are shown in Fig. 1. A B-(W-and X-) configuration is augmented if it contains the probably missing edges (u, v), (u, x), (x, y), (y, v) and also (u, x ), (v, y ), (x , y ) for augmented W-configurations.
In the augmentations the edges cross in the inner face of a X-configuration and in the outer face of a B-configuration. A W-configuration comprises both.
Note that the type of a configuration depends on the embedding and the choice of the outer face or the routing of the base edge, which is drawn black and bold in Fig. 1. A B-configuration becomes an X-configuration if the inner and outer faces are exchanged, and vice-versa. In a W-configuration the roles of the straight-line and curved crossing edges swap by this exchange. This observation was used by Alam et al. [1] in their normal form theorem for embeddings of 3-connected 1-planar graphs. Here, a given embedded 1-planar graph is first augmented by planar edges to a planar maximal 1-planar graph and then the embedding is transformed into normal form by local changes in the cyclic order of the neighbors of some vertices. We recall this result and stress the proof. This triangulates the faces at crossing points. Thereafter, triangulate the planar faces. These steps take linear time, starting from E(G). The result is the embedded supergraph H. Now all B-, W-, and X-configurations are augmented. Each augmented B-configuration which is not a W-configuration is transformed into a X-configuration by the re-routing of the base. Two B-configurations on opposite sides of the base and connected by an edge crossing the base are merged to a W-configuration. There is no vertex inside the boundaries of the end vertices of an X-configuration. E(H) cannot contain two augmented W-configurations or a W-configuration in its interior, since the base of a W-configuration is a separation pair, which is excluded by 3-connectivity. Hence, E(H) is planar maximal by the augmentation and triangulation.
The normal form theorem holds for every 3-connected component of a 1planar graph G. Suppose that G is 2-connected with an embedding E(G) with planar maximal 3-connected components in normal form. For every separation pair {u, v} there is a sequence of 3-connected 1-planar graphs C 0 , . . . , C k−1 in clockwise order at u, and each pair of adjacent components C i and C i+1 with 0 ≤ i ≤ k − 1 is separated by a pair of crossing edges from a W-or an Xconfiguration or both. Otherwise, such components merge to a single planar maximal 3-connected component. If the separation pair is in the outer face and there is no a pair of crossing edges from a W-configuration in the outer face, then the outermost copy e k can be saved.
To separate the components at a separation pair {u.v} even further we allow multiple edges and introduce copies e i for i = 1, . . . , k of the edge e 0 = (u, v) as separation edges. The i-th separation edge e i is routed next to a pair of crossing edges which separates C i−1 from C i .If present the outermost separation edge e k encloses all components and the multi-edges e 0 , e k form the outer face. This situation also holds relative to a separation pair.
For a counting argument each separation edge can be taken for an edge between the components it separates or from a crossing point to a vertex or another crossing point in the planarization.
All steps for the augmentation with 3-connected components in normal form and separation edges take linear time on E(G). Thus we can state. This situation is depicted in Fig. 4, where the copies are the edge e 0 = {u, v} is drawn dotted and blue. Hong et al. [25] showed that a 1-planar embedding can be transformed into a straight-line 1-planar drawing, which preserves the embedding, provided there are no B-and W-configurations. Their algorithm is quite complex and uses the SPQR-tree data structure for the decomposition of the graph into its 3-connected components and the convex drawing algorithm for planar graphs from [9], which needs a high resolution for its numerical computations. There is no stated bound on the area, but it is likely to be exponential. However, each augmented B-and W-configuration induces one edge with a bend if the embedding is preserved. Hence, a straight-line drawing of a 1-planar graph may have a linear number of edges with a bend. The sparse maximal 1-planar graphs from Fig. 3 in [8] may serve as an example.
Here we capture all 1-planar graphs and provide a 1-visibility representation with straight vertical lines for all edges.

Visibility Representation
In this section we show that every 1-planar graph has a 1-visibility drawing. The result is obtained by the 1-VISIBILITY algorithm, whose input is an embedding E(G) as a witness for 1-planarity. After a planar maximal augmentation it considers each 3-connected component C, transformes C into normal form, and separates 3-connected components at a separation pair by separation edges. Then the graph and in particular each 3-connected component is planarized by the extraction of the pairs of crossing edges. The normal form and the separation edges guarantee that each face has at most one pair of crossing edges. The so obtained planar graph is drawn by the common planar visibility algorithm. Thereafter, CROSSING-INSERTION reinserts each pair of crossing edges in the face from which is was extracted. Finally, the edge-segments of added edges are hidden.
Consider a planar visibility algorithm from [14,33,40]. It takes an embedded planar graph and two vertices s, t in the outer faces and directs the edges according to an st-numbering from s to t. Thereafter each vertex v except s, t has a neighbor with a smaller and a larger st-number than itself and two subsequences of incoming and outgoing edges. In other words, each vertex and G are bi-modal [33]. Route the edge (s, t) to the left of the drawing of G. Then consider the directed dual G * , where s * is the face to the right of the (s, t) edge (or the left half of the outer face) and t * is (the right half of) the outer face, and direct its edges according to the s * t * -numbering of G * . Recall that G was extended by separation edges between 3-connected components, which has an impact on G * .
Define the distance δ(v) of a vertex v by its st-number as in [33,40] or for a more compact drawing [14] by the length of a longest path from s and accordingly define the dual distance δ * (f ) of a face f in G * . Then δ(s) = 0, δ(t) = h − 1, δ * (s * ) = 0 and δ * (t * ) = w − 1 for some h ≤ n and w ≤ 2n − 5 and the visibility representation is of size w × h. The insertion of separation edges does not affect the upper bound of 2n − 5, since for each separation edge e i there is at least one edge missing from C i to the next component C i+1 in cyclic order. For the compacted version one must take care that the distance is different for vertices b and d of a quadrangle f = (a, b, c, d), whose bottom and top are a and c if there is an augmented X-configuration. The requirements are met by the st-number and can otherwise be achieved by a local lifting as in [5]. Moreover, if {u, v} is a separation pair with a sequence of 3-connected components C 0 , . . . , C k−1 in clockwise order at u and separation edges e 0 , . . . , e k and the st-number of u is smaller than the st-number of v, then the st-numbering implies that δ(u) < δ(w) < δ(v) for every vertex w from any component C i and δ * (e i−1 ) < δ * (f ) < δ * (e i ) if f is an inner face of C i−1 and δ * (e i ) is the dual distance of the face immediately to the left of e i .
For each edge e = (u, v) let lef t(e) (right(e)) be the dual distance δ * (f ) of the face f of G to the left (right) of v and let lef t(v) (right(v)) be the least (largest) dual distance of a face incident with v.

Algorithm 1: PLANAR-VISIBILITY
Input: A 2-connected planar graph G (with multi-edges) with a planar embedding E(G). Output: A visibility representation VR(G). 1 Construct an st-numbering of G with (s, t) on the left. 2 Compute the dual graph G * . 3 Compute the distance δ(v) for all vertices v of G and the dual distance δ * (f ) for all faces f .
We use PLANAR-VISIBILITY to draw 3-connected components C i of 1planar graphs, whose pairs of crossing edges (a, c) and (b, d) are first extracted and are then reinserted in the face they left behind. The normal form embedding and the added separation edge e i to the right of C i guarantee that each pair of crossing edges has its own face f , which is a quadrangle. f comes from an augmented X-configuration if it is an inner face or is the relative outer face of a W-configuration and is immediately to the left of e i , where e i is a separation edge.
For a face f = (a, b, c, d) let a be the lowest vertex in the visibility drawing of PLANAR-VISIBILITY, i.e., the y-coordinate δ(a) is minimal. We call f a left-wing (right-wing) if δ(a) < δ(b) < δ(c) < δ(d) and b, c are to the left (right) of f , and a diamond if δ(a) < δ(b), δ(d) < δ(c). f is a left-wing if f is the outer face or if (a, d) is a separation edge.
There are always two options, which of the two middle vertices of a quadrangle f is crossed by an edge. A maximal bipartite matching determines one vertex per face and guarantees that each vertex is crossed at most once.
The crossing insertions are illustrated in Fig. 5.  symmetric. Then the edge-segment of (a, d) is at or to the left of δ * (f )−1, and the edge-segments (a, b), (b, c), (c, d) are right aligned at δ * (f ). The vertex-segments of b and c begin at δ * (f ). If f is a diamond with b on the left and d on the right, then β(b) ends at δ * (f ) − 1 and β(d) begins at δ * (f ), and the y-coordinates of b and d are different, since the distance δ guarantees this property. Again there is a single vertex-edge crossing in f . The vertex-segments of the extreme vertices cover the range from δ * (f ) − 1 to δ * (f ), and generally go far beyond.
Finally, consider a separation pair {u, v} and its 3-connected C 0 , . . . , C k−1 , which are separated by separation edges e 1 , . . . , e k as copies of e 0 = (u, v). Associate e i with C i as its base. Then the 3-connected components are sandwiched between the vertex-segments of u and v and two adjacent components C i−1 and C i are clearly separated by e i in a left-to-right order, which is due to the st-and s * t * -numberings.

Algorithm 3: 1-VISIBILITY
Input: An embedded 2-connected 1-planar graph E(G). Output: A 1-visibility representation VR(G) on a grid of quadratic size. 1 Augment E(G) to a planar maximal 1-planar embedding E(G ), e.g., via a maximal planar augmentation of its planarization which keeps the crossing points at degree four. 2 Decompose G into its 3-connected components. 3 foreach separating pair {u, v} do 4 compute the sequence of the 3-connected components Ci for i = 1, . . . , k and add a copy ei of (u, v) as a separation edge to the right of Ci−1. 5 If the embedded graph has a crossing in the outer face, then add a copy of the base edge as a separation edge to cover the crossing from the outer face. Let G be the intermediate graph. 6 Transform the embedding of each 3-connected component of G into normal form. 7 Planarize E(G ) to σ(E(G )) by the extraction of all pairs of crossing edges. 8 Construct a visibility representation of σ(E(G )) by PLANAR-VISIBILITY. 9 (Separately for each 3-connected component) Compute the set of crossed vertex-segments by a maximum bipartite matching on the set of faces F including a pair of crossing edges and the set I of inner vertices of the faces of F . 10 Reinsert the crossing edges by CROSSING-INSERTION. 11 Scale all x-coordinates by the factor 4. 12 Ignore or hide the edges from the augmentations to G and G .
We can now establish our main result. Theorem 1. There is a linear time algorithm to construct a 1-visibility representation of an embedded 1-planar graph on a grid of size at most (8n − 20) × (n − 1).
Proof. First consider the case where the graph G is 3-connected. Its embedding is transformed into normal form with all crossings as augmented X-configurations with the exception of at most one crossing in the outer face. Now each crossing of a pair of edges has its own face, where a crossing in the outer face is assigned to the face to the left of the inserted separation edge, and each such face is a quadrangle. This property also holds for 2-connected graphs by the separation edges between 3-connected components. Hence, the planar graph after the extraction of all pairs of crossing edges can be drawn by PLANAR-VISIBILITY, and the extracted edges can be reinserted by CROSSING-INSERTION. This induces the crossing of a single vertex-edge pair for each pair of crossing edges in f , as shown in Lemma 3, such that each edge is crossed at most once.
Multiple vertex crossings are excluded by a maximum matching between the set of faces F with a crossing and the set of inner vertices I associated with the faces of F . By the st-numbering each vertex v is an inner vertex of at most two faces, one to the left and one to the right. v can be the top or bottom vertex of other faces. Hence, v is assigned to at most two faces of F , and each f ∈ F has two inner vertices, as can be seen from the left-wing, right-wing or diamond shape. The maximum bipartite matching problem over F and I has a solution by Hall's marriage theorem [24], since for every subset F ⊆ F the number of inner vertices |I | of the faces from F is greater or equal to |F |.
In this particular case, a maximum matching can be computed in linear time by first matching all inner vertices of degree one, and then matching the remaining faces using at most one alternation. Since the remaining faces and inner vertices all have degree two, the bipartite graph decomposes into disjoint alternating cycles.
PLANAR-VISIBILITY computes grid points for the segments and uses an area of at most (2n − 5) × (n − 1) including the separation edges. The number of faces of the augmented graph G is bounded from above by 2n − 4, since for each separation edge there is a missing edge between the adjacent 3-connected components. CROSSING-INSERTION does not increase the area, but needs a scaling of the x-coordinates by four, which results in an area of at most (8n − 20) × (n − 1).
All steps take linear time. Steps 1-4, 7, 11 and 12 are done on planar graphs. The linear running time of step 6 is from [1] and of step 8 from [14,33,40].
Step 10 takes O(1) time per crossing, and there are at most n − 2 crossings, and step 5 is a single action. Finally, step 9 is shown above. Corollary 1. Every 1-planar graph is a 1-visibility graph.

Density
It is easily seen that 1-visibility graphs of size n have at most 4n − 8 edges, since there are at most 3n − 6 planar edges and at most n − 2 edges which cross a vertex. This is exactly the upper bound of the density of 1-planar graphs.
Corollary 2. A 1-visibility graph of size n has at most 4n − 8 edges.
From Corollary 1 we obtain a new and simple proof for the maximal density of 1-planar graphs, which was proved before in [6,19,30].
Corollary 3. A 1-planar graph of size n has at most 4n − 8 edges.
Surprisingly, there are 1-visible graphs which are not 1-planar, even if they have the maximum of 4n − 8 edges.
Theorem 2. For every n ≥ 7 there are graphs with 4n − 8 edges which are 1-visible and not 1-planar.
For n ≥ 8 construct the graph G n from K 7 -e and add n − 7 vertices and connect each such v i with vertex 3 on the left and with vertex 1 on the right side and with v i−1 and v i−2 on top, where the edge (v i , v i−2 ) crosses v i−1 , as illustrated in Fig. 6.
Since the 1-planar graphs have the subgraph property G n is not 1-planar. 1-planar (1-visible) graphs with 4n − 8 edges are called optimal [7,39]. Note that there are optimal 1-planar graphs only for n = 8 and n ≥ 10 [7,39], whereas there are optimal 1-visible graphs for every n ≥ 7. More 1-visible and not 1-planar graphs can be constructed using the schema of Fig. 7, where the outer frame represents a subgraph with a unique 1-planar embedding as in [28] and the edge (a, c) crosses vertex b and would cross at least two edges in every 1-planar drawing.

Conclusion and Perspectives
We introduced 1-visibility drawings as a novel crossing style. If this is restricted to a single crossing per object, then edge-vertex crossings properly extend the edge-edge crossings.
The new crossing style raises several questions.

Acknowledgement
I wish to thank K. Hanauer for providing a 1-visibility drawing of the K 7 -e graph.