Fan-Crossing Free Graphs and Their Relationship to other Beyond-Planar Graphs
FFan-Crossing Free Graphs andTheir Relationship to other Beyond-Planar Graphs (cid:63)
Franz J. Brandenburg
Abstract
A graph is fan-crossing free if it has a drawing in the plane so that each edge iscrossed by independent edges, that is the crossing edges have distinct vertices.On the other hand, it is fan-crossing if the crossing edges have a common vertex,that is they form a fan. Both are prominent examples for beyond-planar graphs.Further well-known beyond-planar classes are the k -planar, k -gap-planar, quasi-planar, and right angle crossing graphs.We use the subdivision, node-to-circle expansion and path-addition oper-ations to distinguish all these graph classes. In particular, we show that the2-subdivision and the node-to-circle expansion of any graph is fan-crossing free,which does not hold for fan-crossing and k -(gap)-planar graphs, respectively.Thereby, we obtain graphs that are fan-crossing free and neither fan-crossingnor k -(gap)-planar.Finally, we show that some graphs have a unique fan-crossing free embed-ding, that there are thinned maximal fan-crossing free graphs, and that therecognition problem for fan-crossing free graphs is NP-complete. Keywords: topological graphs, graph drawing, beyond-planar graphs,fan-crossing, fan-crossing free, graph operations
1. Introduction
We consider graphs G that are simple both in a graph theoretic and in atopological sense. Thus there are no multi-edges or loops, adjacent edges donot cross, and two edges cross at most once in a drawing. Graphs are oftendefined by particular properties of a drawing. The planar graphs, in which edgecrossings are excluded, are the most prominent example. There has been recentinterest in the study of beyond-planar graphs [40, 53, 57], which are generallydefined by drawings with specific restrictions on crossings.A graph is k - planar if it has a drawing in the plane so that each edge iscrossed by at most k edges. It is fan-crossing free if the crossing edges are (cid:63) Supported by the Deutsche Forschungsgemeinschaft (DFG), grant Br835/20-1
Email address: [email protected] (Franz J. Brandenburg)
Preprint submitted to Theoretical Computer Science December 14, 2020 a r X i v : . [ c s . D M ] D ec a) (b) (c) (d)Figure 1: (a) A fan crossing, (b) crossings of independent edges (c) a 2-planar crossing, and(d) a quasi-planar crossing independent, i.e., they have distinct vertices, and fan-crossing if the crossingedges have a common vertex, i.e., they form a fan. A drawing of a graphis k - quasi-planar if k edges do not mutually cross. 3-quasi-planar graphs arecalled quasi-planar, see Fig. 1. The aforementioned graphs can also be definedby first order logic formulas [27] and in terms of an avoidance of (natural andradial) grids [2]. A drawing is k - gap-planar [18] if each crossing is assigned toan edge involved in the crossing so that at most k crossings are assigned toeach edge. Then every 2 k -planar drawing is k -gap-planar [18]. These propertiesare topological. They hold for embeddings, which are equivalence classes oftopologically equivalent drawings. Right angle crossing (RAC) is a geometricproperty, in which the edges are drawn straight line and may cross at a rightangle [39].The classes of k -planar, fan-crossing free, fan-crossing, quasi-planar, k -gap-planar, and right-angle crossing graphs are denoted by k PLANAR , FCF , FAN , QUASI , k GAP , and
RAC , respectively. Then 0
PLANAR = 0
GAP is the classof planar graphs. These and other classes have been studied with different in-tensity and depth. In particular, the density [40, 53], which is the maximalnumber of edges of n -vertex graphs, and the size of the largest complete (bipar-tite) graph [11] are well explored, see Table 1 and Fig. 2. It has been shown thatthe recognition problem is NP-complete for 1-planar [50, 58], fan-crossing (fan-planar) [24], 1-gap-planar [18], and RAC graphs [13], whereas a proof for theNP-completeness is still missing for k -planar, k -gap-planar, fan-crossing free,and quasi-planar graphs if k ≥ n − n -vertexfan-crossing graphs with 5 n −
10 edges [32, 55]. Particular examples for fan-crossing and non-fan-crossing free graphs are K and K p,q with p = 3 , q ≥ K n for n ≥
12 and d -dimensionalcube-connected cycles CCC d for d ≥
11 are fan-crossing free but neither fan-crossing nor 1-gap-planar. In addition, for every k ≥
0, there are graphs that areRAC, and thus fan-crossing free, but not k -gap-planar, and thus not 2 k -planar.Our tools, are the subdivision, node-to-circle expansion and path-addition2perations. An s - subdivision expands an edge into a path of length s + 1. Inother terms the edge has s bends. In an s -subdivision of graph G , each edgeof graph G has a j -subdivision for j ≤ s . For subdivision of G , let s be thenumber of edges. A node-to-circle expansion replaces each vertex v of degree d by a circle of length d , so that each edge incident to v is inherited by a vertexof the circle. A path-addition adds a (long) path between any two vertices ofgraph G . These operations distinguish the above graph classes, so that someare closed and others are not. Even more, some classes are universal for s -subdivision ( s ≤
3) and node-to-circle expansion, respectively, that is the imageof any graph is in the particular class.Many more classes of beyond-planar graphs have been studied, see [40, 53,57], some of which are relevant for this paper. A 1-planar drawing is IC- planar [8, 17, 33] if each vertex is incident to at most one crossed edge. It is fan-planar [20, 24, 55] if it is fan-crossing and excludes crossings of an edge from both sides,called configuration II in [55], see Fig. 5(c). Configuration II is a restriction,since there are fan-crossing graphs that are not fan-planar [32]. However, it hasno impact on the density [32] and the results on fan-planar graphs proved in[11, 20, 24].Fan-crossing free graphs were introduced by Cheong et al. [37]. They focuson the density of ( k -)fan-crossing free graphs. Complete and complete bipartitegraphs were studied by Angelini et al. [11]. The state of the art is as follows,see also [40].1. Every n -vertex fan-crossing free graph has at most 4 n − n − n − K is fan-crossing free, whereas K is not. K p,q is fan-crossing free if andonly if p ≤ p ≤ q ≤ n -vertex fan-crossing graph has at most 5 n −
10 edges and the boundis tight.2. Every fan-crossing graph is quasi-planar.3. The fan-crossing graphs and the 2-planar and 1-gap-planar graphs, respec-tively, are incomparable. 3. The recognition problem for fan-crossing graphs is NP-complete.5. K is fan-crossing, whereas K is not. K p,q is fan-crossing if and only ifmin { p, q } ≤ k -planar, k -gap-planar andfan-crossing graphs are universal for O ( n )-subdivision. The k -planar, k -gap-planar, and fan-crossing graphs are not universal for node-to-circleexpansion and for f ( n )-subdivision if f ( n ) ∈ o ( n ) and f ( n ) ∈ o ( n/ log n )for fan-crossing graphs.2. The fan-crossing free, quasi-planar, k -gap-planar ( k ≥
1) and RAC graphsare closed under path-addition, whereas the fan-crossing graphs and the j -planar graphs ( j ≤
2) are not.3. For every k ≥ k -planar, and k -gap planar, respectively.4. The cliques K and K and some other graphs have a unique fan-crossingfree embedding.5. There are thinned maximal fan-crossing free graphs with only 3 . n − . G that extend the planar graphs and are defined by specific properties of cross-ings in drawings [40]. These classes have some properties in common, suchas(i) every planar graph is in G (ii) the density of graphs in G is linear (up to a poly-logarithmic factor)(iii) G is universal for subdivision. 4losed s -subdiv. closed n2c closedgraph class density subdiv. univ. n2c univ. path-add.planar 3 n − n − n ) + – –1-gap-planar 5 n −
10 + Θ( n ) + – +fan-crossing 5 n −
10 + Ω( n/ log n ) ? – –fan-crossing free 4 n − ≤ . n −
20 + 1 + + +RAC 4 n − Table 1: Properties of some classes of beyond-planar graphs, their density, closure undersubdivision, node-to-circle expansion (n2c), and path-addition, respectively, and universalityfor subdivision and node-to-circle expansion. Here “+” means yes, “ − ” means no, and “ ≤ s ”says that s -subdivisions suffice and s − − ” for “?”.. These properties are fulfilled by the aforementioned “major” beyond-planargraph classes and by almost all graph classes listed in [40]. Graphs that aredefined by generalized visibility representations, such as bar- k [48] and shapevisibility [30, 54, 61] satisfy (i)-(iii), as well as graphs with bounded thickness[42, 54], bounded stack number (or book thickness k ) [21, 44, 67] bounded queuenumber [44, 43], mixed stack queue graphs [44], and multiple deque layouts [14].In all these cases, the graphs have a visual representation, which may use anedge coloring and a linear vertex ordering.The properties exclude graphs with bounded genus and minor-closed graphclasses [41], since they fail on (iii), whereas the sets of all graphs and of all non-planar graphs are excluded by (ii). From the collection of graph classes listedin the beyond-planar survey [40], skewness- k and apex- k -graphs are excluded,which are such that the removal of at most k edges and vertices, respectively,makes a planar graph.The closure properties of major classes of beyond-planar graphs are summa-rized in Table 1. Inclusion relations and the containment of special graphs aredisplayed in Fig. 2.The rest of the paper is structured as follows: In Section 2, we study threegraph operations and introduce universality of a graph class for an operation. InSection 3, we establish incomparabilities between some classes of beyond-planargraphs that are unknown so far. Finally, in Section 4, we study properties offan-crossing free graphs and show that some graphs have a unique fan-crossingfree embedding, that there are thinned maximal fan-crossing free graphs, andthat the recognition problem is NP-complete.
2. Graph Operations and Universality
Planar subgraphs and planarization play an important role for the studyof many classes of beyond-planar graphs [40]. This may indicate that beyond-5
UASI K K RAC PLANAR K FAN GAP K K K K K K K K K n K K K CCC T s ( K ) s ( K ) FCF PLANAR K K K K Figure 2: The inclusion diagram shows proper inclusions and incomparabilities among majorbeyond-planar graph classes, and the containment of complete (bipartite) graphs [11]. It isopen, whether there are graphs in
FCF and 1
GAP , respectively, that are not in
QUASI . Wecontribute cube-connected cycles, in particular
CCC , 2- and 3-subdivisions of completegraphs, namely σ ( K ) and σ ( K ), and the tiles-graph T (Thm. 4). A graph is notcontained in a graph class if it is outside its boundary. It may be inside (unlikely) if ittouches the boundary, such as graph T for 2 PLANAR and 1
GAP . planar graphs are close to planar graphs, i.e., they are “nearly planar” [48].A major distinction comes from subdivisions. Clearly, every graph G has asubdivision that is 1-planar. On the other hand, a graph is planar if and onlyif its subdivision is planar. Definition 1.
A class of graphs G is universal for a graph operation f , or simply f -universal, if for every graph G there is a graph G (cid:48) ∈ G such that G (cid:48) = f ( G ) . Clearly, the removal of all edges transforms any graph into a discrete graph,so that the set of discrete graphs is universal for edge removal. Similarly, the setof complete graphs is universal for edge insertion. A graph class G (cid:48) is universalfor f (cid:48) if G ⊆ G ’, f (cid:48) extends f , and G is universal for f .Universality provides a new perspective on the set of graphs. Every graph isprojected into a special class G by an operation f , so that GRAPHS = f − ( G ),where GRAPHS denotes the set of simple graphs. Thus all graphs look likegraphs from G if they are seen through the lens of f .For our further studies, box-visibility representations of graphs and the num-ber of crossings will be useful. In a box-visibility representation , the vertices ofa graph are represented as boxes, which are axis aligned rectangles. Each edgeis an orthogonal polyline with horizontal and vertical segments between theborders of two boxes, so that segments do not traverse other boxes. Edges inci-dent to the same box do not cross. There is a rectangle visibility representation specialbox-visibility representation if each edge consists of a vertical and a horizontalsegment with a bend to the right in between. It is well-known that every graphhas a special box-visibility representation. The idea dates back to the early1980th and Valiant’s representation of graphs of degree at most four [66]. Sim-ply, place the boxes for the vertices on the main diagonal and route each edge uv by a vertical segment from the top of the box of u and a horizontal segmentto the left side of the box of v . Later studies have focussed on box-visibilityrepresentation with small area and few bends [22, 23, 63]. Box-visibility repre-sentations have been used to show that every graph has a 3-bend RAC drawing[39]. There is an alternative 3-bend RAC drawing, in which the vertices areplaced on a horizontal line and the second and third segments are drawn withslope ± cross-ing number cr ( G ) of any graph G with n vertices and m > n edges is atleast cm /n for some c >
0. This fact is known as
Crossing Lemma . Thecurrently best bound is c = 1 /
29 if m ≥ . n [1]. In consequence, completegraphs K n have Ω( n ) many crossings. The Gay or Harary-Hill conjecture statesthat cr ( K n ) = (cid:98) n (cid:99)(cid:98) n − (cid:99)(cid:98) n − (cid:99)(cid:98) n − (cid:99) , which has been proven for n ≤
12 [62].Hence, any drawing of K has at least 150 crossings. Similarly, there are upperand lower bounds on the crossing numbers of complete bipartite graphs [56], hy-percubes and cube-connected cycles, where a lower bound is d − (9 d + 1)2 d − for the d -dimensional cube-connected cycle [65]. A node-to-circle expansion substitutes each vertex v of degree d by a circle C ( v ) of length d , so that each vertex has degree three and each edge incident to v is inherited by a vertex of the circle. If there is a rotation system with the cyclicordering of the edges incident to v , then this ordering is preserved by the node-to-circle expansion. Thereby edges do not cross if they are incident to verticesof a circle, since graphs are simple. There are inner edges between consecutivevertices of a circle and binding edges between vertices of different circles, whichare one-to-one related to the original edges. A node-to-box expansion is thespecial case, in which the edges of each circle are uncrossed. We denote thenode-to-circle expansion of a graph G by η ( G ), which is a 3-regular (cubic)graph with 2 m vertices and 3 m edges if G has m edges and no vertices of degreeone or two.A node-to-circle expansion is a special split operation, in which each vertexis replaced by a subgraph H , so that the vertices of H inherit all adjacenciesfrom the vertex. In our case, H is a circle. Graph H is a discrete graph in the k -split operation from [47], where the objective is to transform a graph into aplanar one by as few k -split operations as possible.Node-to-circle expansions have been used in VLSI theory to transform ahypercube into a cube-connected cycle [60]. The d -dimensional hypercube H d consists of 2 d vertices denoted by d -digit binary numbers. There is an edge if7 a) (b)Figure 3: A 1-planar drawing of the node-to-circle expansion of (a) K ,n and (b) K . In (b)the topmost circle (striped) intersects two other circles.(a) (b)Figure 4: Quasi-planar drawings of (a) K and (b) η ( K ), where edges from K , drawndashed and red and blue, are drawn dotted and red and blue in η ( K ). These edges arecrossed six times in K , whereas the binding edges in η ( K ) are crossed at most four times,including two crossings on the boundary of the traversed circle, drawn as a big node. and only if the Hamming distance between the binary numbers is one. The d -dimensional cube-connected cycle CCC d is obtained from H d by beveling eachcorner, so that CCC d is the node-to circle expansion of H d , see Fig. 8. It has d d vertices of degree three. Hypercubes and cube-connected cycles have similarproperties, such as the crossing number [65], diameter and separation width [60].The node-to-circle expansion operation preserves some graph properties,such as 3-connectivity and genus, and simplifies others, such as crossings. If G has crossing number k , then η ( G ) has crossing number at most k . For exam-ple, K has 60 crossings and is not 4-planar [11], whereas η ( K ) has at most45 crossings and is 4-planar, as Fig. 4(b) shows. The complete bipartite graphs K ,n are fan-crossing but are not k -planar for k = 1 , , , n = 5 , , , K , is not fan-crossing free. However, thenode-to-circle expansion of K ,n is 1-planar, as Fig. 3(a) illustrates, and it hasfewer crossings than K ,n . Clearly, the crossing numbers of G and η ( G ) coincideif the inner edges of η ( G ) are uncrossed, i.e. in a node-to-box expansion. Lemma 1.
The node-to-circle expansion η ( G ) of a graph G is planar if and e v ev e v e (a) v e v ev e v e (b) v e v ev e v e (c) v e v ev e v e (d)Figure 5: An edge e that is crossed by three edges incident to vertex v in a 3-regular fan-crossing graph. Edge e is crossed from (a) one and (c) both sides (configuration II). A newplacement of v and an edge rerouting towards a 2-planar graph. only if G is planar. Similarly, η ( G ) is fan-crossing free ( k -planar, k -gap-planar,quasi-planar, RAC) if so is G .Proof. Construct graph η ( G ) from a drawing of G . Consider a small box aroundeach vertex, so that two boxes do not intersect and a box is not crossed by anon-incident edge. For each edge e incident to v place a new vertex v e at the lastintersection of e and the box, clip e at v e and remove v . Link two vertices by anedge that are adjacent in the boundary of a box, so that each box is transformedinto a circle with uncrossed edges. For each edge e = uv of G there is an edge e (cid:48) between the boxes or circles of u and v , so that e (cid:48) inherits all crossings from e . Hence, η ( G ) is fan-crossing free ( k -planar, k -gap-planar, quasi-planar, RAC)if so is G . (cid:117)(cid:116) Corollary 1.
The graph classes
FCF , k PLANAR , k GAP , QUASI , and
RAC areclosed under node-to-circle expansion ( k ≥ ). Angelini et al. [10] observed that every fan-planar drawing of a degree-3graph is 3-planar. This fact can be improved. Here configuration II comes intoplay and edges are rerouted as in [32].
Lemma 2.
Graph G is 2-planar and thus 1-gap-planar if it is a fan-crossinggraph of degree at most three.Proof. Consider a fan-crossing drawing of G . If edge e is crossed by three edges,then the crossing edges are incident to a vertex v and e is not crossed by anyother edge. Now vertex v can be moved close to the crossing point of e and themiddle of the three crossing edges. Similar to [32], the edges incident to v arererouted first along e . Then e is uncrossed if it is crossed from one side, as inthe fan-planar case [55], and it is crossed at most once, in general, see Fig. 5.Hence, there is a 2-planar drawing, which is 1-gap-planar [18]. (cid:117)(cid:116) In particular, the node-to-circle expansion η ( G ) is 2-planar and 1-gap-planarif it is fan-crossing. It is unclear whether η ( G ) is fan-crossing if G is fan-crossing. It is true for maximal complete (bipartite) fan-crossings graphs, thatis K and K ,n , where the node-to circle expansions are even 1-planar, as Fig. 3shows. These cases are singular. Consider a 4 × n ≥ a) (b)Figure 6: (a) A special box-visibility representation of K and (b) its node-to-circle expansion.Inner edges are drawn dotted. that there is K ,n including the four corners of each quadrangle. We conjecturethat the node-to-circle of this graph is not fan-crossing. Theorem 1. (i) The fan-crossing-free graphs and the quasi-planar graphs areuniversal for node-to-circle expansion.(ii) The fan-crossing, k -planar and k -gap-planar graphs ( k ≥ ) are notuniversal for node-to-circle expansion.Proof. For a fan-crossing free drawing of η ( G ), consider a drawing of G with allvertices on a circle in the outer face. Replace each vertex v by a circle C ( v ) thatdoes not intersect any other circle or an edge that is not incident to v , so that allinner edges are uncrossed. If two edges of G cross, then so do the correspondingbinding edges, which are independent, so that η ( G ) is fan-crossing free.For quasi-planar graphs, consider a special box-visibility representation of G , in which the boxes are arranged along the diagonal. Each edge consists ofa vertical and a horizontal segment with a bend to the right. Direct it upwardso that each box is incident to incoming edges on the left and of outgoing edgeson top, see Fig. 6(a). For each vertex v construct a circle C ( v ) consisting of thebend points of the outgoing edges and the points of the incoming edges at theleft side of the box of v , as shown in Fig. 6(b). The ordering of C ( v ) coincideswith the rotation system at the box of v . The inner edges from the cycles arealmost vertical. Two such edges do not cross, since they are ordered left to rightaccording to the left to right ordering of the boxes. The binding edges are thehorizontal segments of the edges from the special box-visibility representation.Such edges may cross inner edges. Nevertheless, there are no three mutuallycrossing edges, so that η ( G ) is quasi-planar.For (ii) consider hypercubes and cube-connected cycles as their node-to-circle expansion. The crossing number of CCC d is at least d − (9 d + 1)2 d − [65], so that some edges are crossed at least c d /d times for some c >
0, see[10]. Hence, for every k ≥ k -(gap)-planar and not fan-crossing by Lemma 2. (cid:117)(cid:116) In consequence, the 11-dimensional cube-connected cycle
CCC is fan-crossing free and quasi-planar but not 1-gap-planar. Moreover, it is not 2-planar1018] and not fan-crossing by Lemma 2.Note that the fan-crossing free and quasi-planar drawings of η ( G ) are differ-ent. They are not necessarily quasi-planar and fan-crossing free, respectively.It is open whether every graph G has a drawing of η ( G ) that is simultaneouslyquasi-planar and fan-crossing free, called grid-crossing in [27], or that is RAC.A RAC drawing can be obtained by a generalized version of a node-to-cycleexpansion in which a circle C ( v ) of a degree- d vertex has d vertices of degreethree and two vertices of degree two. Then C ( v ) can be drawn as a box withvertical and two horizontal segments. Moreover, every graph has a generalizednode-to-circle expansion that is outer fan-crossing free, so that all vertices arein the outer face. An s -subdivision of an edge replaces it by a path of length s +1. In geometryand in particular in orthogonal [46] and RAC drawings [39], a subdivision iscalled a bend. Each edge is replaced by a path of length at most s + 1 in an s -subdivision σ s ( G ) of graph G , where s is the number of edge for a subdivisonof G . An s -subdivision is uniform if each edge of graph G is extended to a pathof length s + 1. Then σ s ( G ) is a single graph. In the general version, σ s ( G ) isparameterized by its set of edges and is a set of graphs. Here these cases donot matter, since too long paths can be folded, so that some of its edges remainuncrossed. Subdivisions of graphs are used in Kuratowski’s theorem, whichstates that a planar graph does not contain a subgraph that is a subdivision of K or K , , i.e., K and K , are the topological minors of the planar graphs.Note that edge contractions, i.e., paths of length zero, are generally used in thetheory of graph minors [41].Clearly, the ( s -)subdivision of a graph G is in a graph class G if G is one ofthe aforementioned graph classes. If edges of a fan with a common vertex v arecrossed, then their segments at v are crossed after a subdivision. Lemma 3.
The graph classes
PLANAR , k PLANAR , k GAP , FCF , FAN , QUASI ,and
RAC are closed under s -subdivision for every s ≥ . A graph is
IC-planar [8] if it admits a 1-planar drawing so that each vertexis incident to at most one crossed edge. Structural properties have been studiedin [17]. Brandenburg et al. [33] have shown that IC-planar graphs admit a 1-planar straight-line drawing with right angle crossings. Hence, IC-planar graphsare simultaneously 1-planar and RAC. Hence, a universality of the IC-planargraphs transfers to all graph classes containing them.
Theorem 2. (i) The IC-planar, k -planar, and k -gap-planar ( k ≥ ) graphsare universal for f ( n ) subdivision if and only if f ( n ) ∈ Θ( n ) . The fan-crossing graphs are universal for f ( n ) -subdivision if f ( n ) ∈ O ( n ) . Theyare not g ( n ) -subdivision universal if g ( n ) ∈ o ( n/ log n ) .(ii) The RAC graphs are universal for 3-subdivision and not for 2-subdivision.(iii) The fan-crossing free graphs are universal for 2-subdivision. iv) The quasi-planar graphs are universal for 1-subdivision and not withoutsubdivisions.Proof. Consider a drawing of graph G . Subdivide each edge into segments, sothat each segment is crossed at most once and each subdivision point is incidentto at most one crossed segment. The so obtained drawing is IC-planar. Eachedge of G is crossed at most by all other edges, since G is simple. Thus O ( n )subdivisions per edge suffice. For the lower bound consider the crossing numberof complete graphs K n , which is Ω( n ) [5, 59]. Clearly, the number of crossings isunchanged by subdivisions. Since k -planar and k -gap-planar graphs admit O ( m )crossings, where m is the number of edges, some edges have Ω( n ) crossingsand need Ω( n ) many subdivisions for k -(gap)-planar graphs. Clearly, O ( n )subdivisions suffice for fan-crossing graphs. For the lower bound consider d -dimensional cube-connected cycles. By the lower bound on the crossing number[65], there are edges with at least
130 2 d d − c n log n for some c >
0, since n = d d . By Lemma 2, 3-regular fan-crossing graphsare 2-planar, so that at least cn/ log n many subdivisions are needed for someedges.Didimo et al. [39] have shown that every graph has a 3-bend RAC drawing,whereas n -vertex graphs with a 2-bend RAC drawing have O ( n / ) edges, whichproves (ii).For (iii), every graph can be drawn such that each edge consists of three seg-ments, the first and last of which are uncrossed and the middle segments inheritall crossings. Hence, the drawing is fan-crossing free and has two subdivisions.Finally, for (iv), consider a special box-visibility representation of G . Useeach bend as a subdivision point and draw each vertex v as a point in its box.The incident edges of v are extended from the boundary of the box to v sothat there is no crossing in the interior of a box. Then only horizontal and(almost) vertical segments of edges may cross, so that no three edges cross oneanother. Since n -vertex quasi-planar graphs have at most 6 . n −
20 edges [3],one subdivision is necessary to represent all graphs. (cid:117)(cid:116) A path-addition adds an internally vertex-disjoint path P between any twovertices of a graph. Such paths are also known as ears. They are used inear decompositions of 2-connected graphs [51] and in subdivisions. We wishto use path-additions so that they preserve a given class of graphs. Therefore,the added paths are long. Their length is at least cn for some c > n -vertex graphs. This property distinguishes our path-addition operation fromear decompositions [51]. In comparison with subdivision, a path-addition addsa (long) path between any two vertices, whereas a subdivision can do so if thereis an edge. Hence, path-additions can create a subdivision of K n from anynon-empty graph. A more general version of path-addition was introduced in[36] and studied in [35]. For a graph G and a path P , let G ⊕ P denote thegraph obtained by adding the vertices and edges of P to G , where the internalvertices of P are new and have degree two. A class of graphs G is closed under12ath-addition if there is some function D G ( n ) so that G ⊕ P is in G if G is an n -vertex graph in G and P has length at least D G ( n ).It is readily seen that the graph classes RAC , FCF , k GAP for k ≥
1, and
QUASI are closed under path-addition if D G ( n ) is a linear function. In particular, k -gap-planar graphs are made for path-additions, since the crossings created byan added paths are assigned to its edges if the length of the path exceeds itsnumber of crossings. For a non-closure it must be shown that there are graphsso that the addition of a path of any length violates the defining properties ofthe graph class. Theorem 3.
The graph classes
RAC , FCF , k GAP ( k ≥ ), and QUASI areclosed under path-addition, whereas k PLANAR ( k ≤ ) and FAN are not.Proof.
For the positive closure results, let D G ( n ) be twice the density function.Path P is routed along a (shortest) path S in G . Path P makes a bend andcreates a further subdivision if it crosses an edge incident to an internal vertexof S , so that the crossing introduces a violation without the subdivision. In aRAC drawing, the edges of P can be drawn so that they cross an edge of G at aright angle. If a graph is disconnected and the first and last vertices of P are indifferent components, then there are exits, which are vertices or crossing pointsin the outer face of a component, so that P can be routed along such exists.Each bend is charged to an edge of the given graph, which is charged at mosttwice. If an edge of G crosses P multiple times, then it crosses S so that thesecrossings are assigned to the edges of S .For the negative results, consider a 4 × G with vertices ( x, y )for 1 ≤ x, y ≤
4. Triangulate G by edges parallel to the diagonal. Then replaceeach edge by a “fat edge”. For fan-crossing (fan-planar) graphs, a fat edge uv is K with vertices u and v in the outer face of an embedding. Call the resultinggraph H . Binucci et al. [24] have shown that K has an almost unique fan-planarembedding with two vertices in the outer face, which results from fragments ofcrossed edges, see Fig. 10(c). The result holds for fan-crossing graphs, sinceconfiguration II [55] is not used, and for 2-planar graphs. Fat edges cannotbe crossed without violation. In case of 1-planar graphs, a fat edge is K , seeFig. 10, and an edge for planar graphs. In consequence, any fan-crossing or 1-or 2-planar embedding of H is a grid with fat edges. For every fan-crossing( j -planar with j ≤
2) drawing of H , there is a fat edge between the vertices u and v at positions (2 ,
2) and (3 , P cannot be added between u and v so that H ⊕ P is fan-crossing( j -planar with j ≤ G is fan-crossing ( j -planar). (cid:117)(cid:116) Hence, the path-addition operation distinguishes the graph classes
RAC , FCF , k GAP ( k ≥ QUASI from
FAN , 1
PLANAR and 2
PLANAR , respectively.Since every 1-planar graph is fan-crossing free [37], every 2-planar graph is 1-gap-planar [18], and every fan-crossing or 2-planar graph [9] is quasi-planar, itprovides an alternative proof for the properness of these inclusions.13 igure 7: An internally crossed 5 ×
3. Relationships
The following proper inclusions are known: Every 2 k -planar graph is k -gap planar, which in turn is 2 k +2-quasi-planar [18]. Every RAC graph is fan-crossing free and quasi-planar [39]. Every fan-crossing graph is quasi-planar[37]. In addition, the following incomparabilities are known: (i) 1-planar andRAC [45] (ii) 2-planar and fan-crossing (fan-planar) [24], and (iii) fan-crossingand 1-gap-planar, where the latter, stated as an open problem in [18], followsfrom K and K ,n for n ≥ Theorem 4.
There are RAC graphs that are not fan-crossing.Proof. A tile T is an internally crossed 5 × t p,q for 1 ≤ p, q ≤ t p,q t r,s if max {| p − r | , | q − s |} = 1. It has 25 vertices and 72 edgesand diameter four, see Fig. 7.Since outer fan-planar graphs have a density of 3 n − T has at least one vertex in its interior, called the center of T , andat most 24 edges in the boundary of its outer face. There are at least threevertex-disjoint paths between the center and vertices in the boundary, since atile is 3-connected.Let graph G consists of at least 650 tiles and at least 60 paths between anytwo vertices from different tiles. Suppose there is a fan-crossing embedding of G . Each tile has a Jordan curve for its boundary, so that the center is in theinterior of the Jordan curve. We say that two tiles cross, nest, and are disjointif their boundaries cross, nest or are disjoint, respectively. They nest if one tileis in the interior of the boundary of the other tile. Define the distance betweentwo tiles by the minimum number of boundaries that must be crossed by a path P between their centers if P is added to G in a fan-crossing embedding of G ⊕ P .In particular, the distance is at least two if the tiles are disjoint, since an addedpath must cross the boundary of each of them.14uppose there are (at least) five tiles T i with mutual distance at least two.Then consider p ≥
60 paths of length 15 between their centers. Each pathmust cross at least two boundaries. Since each boundary consists of at most 24edges, at least one of its edges is crossed twice if there are at least 25 paths.In a fan-crossing embedding, this edge is crossed by the first (or last) edges ofthe paths, since all other edges of the paths are independent. We call P a redpath if the first and last edges cross an edge of a boundary. Suppose there is ared path between the centers of tiles T i and T j for 1 ≤ i, j ≤
5. Then there isa subgraph homeomorphic to K , so that at least two paths must cross. Sincethe paths have length 15 and their first and last edges are crossed by edges ofboundaries, at least one inner edge of a path is crossed twice by independentedges from two paths if there are at least 14 parallel red paths. Then there is aviolation of a fan-crossing embedding.Since the centers of the tiles are unknown, there are p ≥
60 paths of length15 between any two vertices from different tiles. Graph G is not fan-crossingif there are at least five tiles at mutual distance two. Since the boundary of atile has length 24, at least 60 −
23 paths from the center cross the boundary infans for size at least two, and at least 60 −
46 paths between two centers arered. Each red path has 13 edges between the boundaries, so that there is anindependent crossing if there are at least 14 red paths between the centers offive tiles.Graph G is a RAC graph. Therefore, draw each tile as a RAC graph asshown in Fig. 7, so that the edges between grid points are parallel to the axis.Place the tiles from left to right so that they are disjoint and their bottom rowis on the x-axis. There is a bundle of paths between any two vertices fromdifferent tiles. Route each added path P from a vertex to the outer face of thetile, so that internally there are right angle crossings. This needs paths of lengthat most six, so that the sixth edge is parallel to the y -axis. The middle edgesbetween the seventh and tenth vertex of each path are horizontal or verticaland are routed outside the tiles, so that edges of two paths do not overlap. Ifthey cross, they cross at a right angle. If there are several paths between twovertices, then they are piecewise in parallel, except for the first and last edges.It remains to show that there is a graph consisting of at least five tiles atdistance at least two. Consider a fan-crossing embedding E ( G ) of k tiles, so thatonly the outer boundary and the center in its interior are taken into account.In other words, there are k Jordan curves and k points, so that a point is inthe interior of a particular Jordan curve. The boundary of a tile can be crossedat most 24 times by the boundaries of other tiles, since it has length at most24. In fact, three or four edges from both boundaries are involved in a crossing.Consider the planar dual of E ( G ). If center c of tile T is in face f , then assign T to f . Face f has at most 25 adjacent faces, that are accounted to f , sincethe boundary of a tile must be crossed for a new face and it can be crossed atmost 24 times. In addition, there is the outer face or the face from a nestingtile enclosing T . Faces from tiles in its interior are accounted to these tiles.Moreover, at most five tiles can be assigned to a face, since a tile has diameterfour and any path from the center to a vertex in the boundary of T can cross at15ost four boundaries. Suppose there are at least 650 tiles. If several tiles areassigned to a face, then keep only one them. Each face has at most 25 neighborswith an assigned tile at distance one, so that there are 26 candidates. Hence,at least five tiles remain, so that the faces, to which they are assigned, havedistance at least two. (cid:117)(cid:116) There should be simpler counterexamples, for example, a 10 ×
10 tile with anadded path between the vertices at positions (4 ,
4) to (7 , Theorem 5.
If the crossing number exceeds the number m of edges of graph G so that cr ( G ) > km , then the 3-subdivision σ ( G ) is a RAC graph, and thusfan-crossing free and quasi-planar, whereas σ ( G ) is not k -gap-planar and not k -planar.Proof. Graph σ ( G ) is a RAC graph by Theorem 2(ii). It has at most 3 m edges, so that there are at most 3 km crossings in a k -gap-planar drawing of σ ( G ). Since the crossing number is preserved by subdivisions, σ ( G ) is not k -gap-planar if cr ( G ) > km . Then it is not 2 k -planar by [18]. (cid:117)(cid:116) Bae et al. [18] have shown that the number of edges in n -vertex k -planargraphs is at most max { . √ k, . } n and at most 5 n −
10 for k = 1. TheCrossing Lemma and cr ( K ) = 150 [62] provide simple examples of non 1-gap-planar graphs. Our example is an alternative to the RAC and non 1-planargraph by Eades and Liotta [45], which has 85 vertices and is constructed from K by an addition of many short paths similar to the graph in Theorem 4. Corollary 2.
The 3-subdivision of K is RAC and not 1-gap-planar (2-planar).The 2-subdivision of K is fan-crossing free and not 1-gap-planar (2-planar) Angelini et al. [10] observed that cube-connected cycles are not k -planar.Their arguments apply to k -gap-planar and fan-crossing graphs, so that weobtain alternative proofs for quasi-planar graphs that are neither k -gap-planar[18] nor fan-crossing [11, 3, 24]. Theorem 6.
For every k , there are cube-connected cycles that are fan-crossingfree and quasi-planar, respectively, and are not k -planar, k -gap planar, and fan-crossing, respectivelyProof. Cube-connected cycles are fan-crossing free and quasi-planar by Theo-rem 1. They are not k -(gap)-planar if the crossing number [65] exceeds k -timesthe number of edges, which holds for d -dimensional cube-connected cycles and k ≤
130 2 d d −
4. For fan-crossing graphs use Lemma 2. (cid:117)(cid:116)
Corollary 3.
CCC is fan-crossing free and quasi-planar but neither fan-crossing nor 3-gap-planar nor 6-planar. UASI K K RAC PLANAR K FAN GAP K K K K K K K K K n K K K CCC T s ( K ) s ( K ) FCF PLANAR K K K K Figure 8: 1-planar drawings of H and CCC . Proof.
The lower bound for the crossing number of
CCC [65] exceeds thenumber of edges of CCC by a factor c > .
1. Hence, it is not 3-gap-planarand thus not 6-planar [18] and not fan-crossing by Lemma 2. (cid:117)(cid:116)
Note that the cube H is planar so that CCC is planar. Figure 8 showsa 1-planar drawing of H , which can be transformed into a RAC drawing, andan IC-planar drawing CCC , so that these graphs are 1-planar and RAC [33].Every cube-connected cycle is 1-bend RAC [12], that is σ ( CCC d ) is RAC. Wecan summarize: Corollary 4.
Any two graph classes from (a)-(c) are incomparable: (a) fan-crossing free and RAC, (b) fan-crossing, and (c) 1-gap-planar and 2-planar.
Figure 2 displays the relationships between major beyond-planar graph classes.The shown inclusions were known, as well as the incomparability between fan-crossing and 2-planar resp. 1-gap planar graphs. We have added incompara-bilities for the fan-crossing-free and RAC graphs. Still open is the relationshipbetween the quasi-planar graphs and the fan-crossing free resp. 1-gap-planargraphs. We conjecture an incomparability.
4. Properties of Fan-Crossing Free Graphs
It is well-known that the 4-clique K admits two embeddings, as a planartetrahedron and 1-planar with a pair of crossing edges. The 5-clique K hasfive embeddings [52], as shown in Fig. 9. Only the so-called T-embedding inFig. 9(a) is fan-crossing free and even 1-planar. The embedding in Fig. 9(e)has an edge which is crossed by the edges of triangle. It is 1-gap-planar andquasi-planar and not fan-crossing and not 2-planar. In fan-crossing embeddingsit can be transformed into the Q-embedding of Fig. 9(c) [32].17 a) (b) (c) (d) (e)Figure 9: All non-isomorphic embeddings of K [52], each with two drawings. Only theT-embedding (a) is 1-planar and fan-crossing free.(a) (b) (c)Figure 10: The fan-crossing free embedding of K drawn (a) with a triangle and (b) withonly two vertices in the outer face. (c) A 2-planar, fan-crossing embedding of K . Edges arecolored black, blue, and red if they are uncrossed and crossed once and twice, respectively. Consider a fan-crossing free embedding of K , which is obtained by placingthe next vertex into one of the faces of the T-embedding of K . There are twopossibilities up to symmetry: in a face with or without a crossing point. Onlythe latter results in a fan-crossing free embedding. The obtained embedding isunique, since the edges must be routed as shown in Fig. 10. Otherwise, an edgeis crossed by at least two edges of a fan. The embedding can be drawn with twoor three vertices in the outer face. The 7-clique is not fan-crossing free, sinceit has too many edges, but fan-crossing (fan-planar) [24], see Fig. 10(c). Wesummarize: Lemma 4.
The cliques K and K are fan-crossing free and each has a uniquefan-crossing free embedding. Cheong et al. [37] have shown that every fan-crossing free embedding of an n -vertex graph with 4 n − optimal 1-planar [25]. The embedding consists of a 3-connected planar sub-graph, in which each face is a quadrangle with a pair of crossing edges. Suchgraphs exist for n = 8 and for all n ≥
10 [25]. They have a unique 1-planarembedding, except for extended wheel graphs XW k for k ≥
3, which havetwo embeddings, where the poles are exchanged [64]. An extended wheel graph XW k consists of two poles p and q and a circle of length 2 k . There is an edgebetween each pole and each vertex of the circle, whereas there is no edge pq .18 igure 11: A fan-crossing free embedding of a nested triangle graph, augmented by “hermits”,drawn by pink squares, for the proof of Theorem 7. The drawing is not quasi-planar. A re-routing of the pink edges makes it quasi-planar. In addition, there is an edge between a vertex of the circle and the vertex afternext (in cyclic order). Each of these edges is crossed by an edge incident to apole. Note that optimal 1-planar graphs can be recognized in linear time [29].
Corollary 5.
Every n -vertex fan-crossing free graph with n − edges has aunique embedding, except for the extended wheel graphs XW k , which have twoembeddings. The crossed nested triangle graph ∆ k consists of k nested triangles T , . . . , T k with vertices a i , b i , c i for i = 1 , . . . , k , so that each quadrangle between twosuccessive sides of the triangles has a pair of crossing edges and is K , as shownin Fig. 11. Triangle T i is at level i , where T is in the outer face. The subgraphinduced by two consecutive triangles is K , which is an inner K if it is inducedby T i , T i +1 for i = 2 , . . . , k −
2. Graph ∆ k has 3 k vertices and 12 k − K subgraphs. Lemma 5.
For every k ≥ , the crossed nested triangle graph ∆ k has a uniquefan-crossing free embedding.Proof. For k = 1 there is a triangle, and K for k = 2 has a unique fan-crossingfree embedding by Lemma 4. For k ≥
3, the restriction of ∆ k to two consecutivetriangles is K . Each such K has a unique fan-crossing free embedding, so19hat it is drawn with a triangle in its outer face. Otherwise, there is a W-configuration, as shown in Fig. 10(b), which enforces a crossing of an edge byat least two edges of a fan with the common vertex in K and the other verticeson the previous or next level. Hence, all triangles are drawn as nested triangles.Clearly, one can choose the outer face of the drawing with three vertices or withtwo vertices and a crossing point. (cid:117)(cid:116) Note that the crossed nested triangle graph ∆ k admits many fan-crossingdrawings, since a K has many fan-crossing embeddings with different embed-dings of its K subgraphs.Next we consider a traversal of K or K by a path in a fan-crossing freeembedding, which causes a delay by at least three bends. This is used in theNP-hardness proof of Theorem 8. Lemma 6.
In every fan-crossing free embedding, if path P traverses K ( K ) ,so that P and the clique have disjoint sets of vertices and there are differentedges of K ( K ) crossed first and last by P , then at least four (five) edges of P cross edges of K ( K ) .Proof. Let P = ( u , . . . , u t ) be a path such that u and u t are in the outer faceand u , . . . , u t − are in an inner triangle of the T -embedding of K . Let the firstedge ( u , u ) of P cross edge ( v , v ) of K and let the last edge ( u t − , u t ) of P cross ( v , v ) in a fan-crossing free embedding of a graph G that includes P and K . Since K has a unique fan-crossing free embedding, P must cross twomore edges of K , since a simultaneous crossing of two or more edges induces afan-crossing. Hence, P needs at least four edges for a traversal of K . The caseof K is similar. (cid:117)(cid:116) A graph G is maximal for a class of graphs if no edge can be added withoutviolation, so that G + e is not in the class. The density is the maximum numberof edges of all maximal n -vertex graphs in a class. The sparsity is the minimum.Density and sparsity coincide for planar graphs, whereas they differ for someclasses of beyond-planar graphs. Brandenburg et al. [34] have shown that thesparsity of 1-planar graphs is at most n − . Such graphs are obtained by socalled hermits , which are vertices of degree two that cannot be linked to othervertices without violation. Similar results have been obtained for 2-planar [34],IC, NIC and outer 1-planar graphs [15, 17]. Theorem 7.
There are maximal fan-crossing free graphs with m = 7 / n − / edges for every n = 6 k + 1 with k ≥ .Proof. Consider a crossed nested triangle graph ∆ k for k ≥
3. Attach threehermits h i , h (cid:48) i and h (cid:48)(cid:48) i along the edges a i a i +1 , b i b i +1 and c i c i +1 of each triangle T i = ( a i , b i , c i ) for i = 1 , . . . , k − h i h (cid:48) i , h (cid:48) i h (cid:48)(cid:48) i and20 a) (b)Figure 12: Illustration to the proof of Theorem 7 if hermits are moved to other faces. h i h (cid:48)(cid:48) i . Add a hermit in the outer face and link it to a , b , c , and similarly forthe inner face, see Fig. 11.Graph H k is fan-crossing free, as shown by Fig. 11, where the embeddingis not quasi-planar. We claim that it is maximal fan-crossing free. Therefore,observe that ∆ k is maximal fan-crossing free and 1-planar and has a unique fan-crossing free embedding. Consider hermit h i that is attached to edge a i a i +1 .Then h i can be placed into the faces f and f (cid:48) to either side of a i a i +1 . It cannotbe placed into another face without creating a fan-crossing by two edges. Also h i cannot be linked to another vertex of ∆ k without creating a crossing by a fanof at least two edges. The case of h (cid:48) i and h (cid:48)(cid:48) i is similar. Hermit h i is linked to h (cid:48) i and h (cid:48)(cid:48) i by an edge, but not to any other hermit if the embedding is fan-crossingfree. Then at least two edges incident to a vertex of ∆ k must be crossed. Ifthree hermits h i , h (cid:48) i , h (cid:48)(cid:48) i are simultaneously placed into another face that is nextto the one in which they were placed, then the edges between them create acrossing by a fan of two edges, as illustrated in Fig. 12(a). Similarly, the hermitin the inner face can be placed in a neighboring face, as illustrated in Fig. 12(b),but it cannot be linked to another hermit or another vertex without violation.Hence, the fan-crossing free embedding of H k is unique, as claimed.Graph ∆ k has 3 k vertices and 12 k − k −
1) + 2hermits and 9( k −
1) + 6 edges incident to the hermits. Hence, graph H k has6 k − k −
12 edges, so that m = 7 / n − / (cid:117)(cid:116) The recognition of beyond-planar graphs is mostly NP-hard, see [40]. It hasbeen expected that recognizing fan-crossing free graphs is NP-complete, too,which has been stated as an open problem in [40] and is proved next.
Theorem 8.
The recognition of fan-crossing free graphs is NP-complete.Proof.
Clearly, the problem is in NP.For the NP-hardness we adapt the construction given by Grigoriev and Bod-laender [50] for 1-planar graphs. The proof is by reduction from 3-PARTITION,which is a strongly NP-hard problem [49]: An instance I of 3-PARTITIONconsists of a multiset A of 3 m positive integers with B/ < a < B/ (cid:80) a ∈ A = mB for some integer B and a ∈ A . The 3-PARTITION problem asks21hether A can be partitioned into m subsets A , . . . , A m , each of size three,such that the sum of the numbers in each subset equals B .From I we construct a graph G I in polynomial time, so that I has a solutionif and only if G I admits a fan-crossing free embedding. The components of thereduction are a transmitter, a collector, m splitter and “fat” edges. A fat edge uv is a K with vertices u and v and three more vertices, where u and v arein the outer face if there is a fan-crossing free drawing. For an illustration seeFig. 13.The transmitter is a double-wheel with a center c T and two circuits of length3 m . Let C = ( u , . . . , u m ) be the outer and C (cid:48) = ( u (cid:48) , . . . , u (cid:48) m ) the innercircuit. Let c T u (cid:48) ) and u (cid:48) i u i be fat edges for each i . Then the center c T is a vertexof each of the K graphs of the fat edges that are incident to the center. Eachsector with boundary u i , u (cid:48) i , c T , u (cid:48) i +1 , u i +1 is partitioned into three triangles by(normal) edges u (cid:48) i u (cid:48) i +1 , u i u i +1 and a diagonal u i u (cid:48) i +1 . The triangle ( u i , u (cid:48) i , u (cid:48) i +1 )is called a sector-triangle , and ( u i , u i +1 , u (cid:48) i +1 ) is called an outer triangle , see theenlargement in Fig. 13(b). The collector is defined accordingly, with a center c C and Bm sectors with an outer boundary ( v , . . . , v Bm ). The boundary of thetransmitter is partitioned into m segments, each of width three, and accordingly,the collector has m segments, each of width B . The ends of the segments areconnected by a fat edge in circular order. Hence, there is fat edge ( u i , v Bi ) for i = 1 , . . . , m .For each element of A there is a splitter P a with center c a and a + 1 satellitesat distance two from the center, i.e., each satellite has a connector on the pathto c a . In [50], the satellites are at distance one from the center. We connectone satellite of each splitter with the transmitter center c T and the remainingsatellites with the collector center c C . Hence, there is a path of length six from c T via each c a to c C .Since 3-PARTITION is strongly NP-hard, all numbers a ∈ A can be givenin unary encoding, such that G I has polynomial size and can be constructed inpolynomial time.For the correctness of the reduction we follow the arguments given in [50].However, we use K instead of K as a fat edge, and sector triangles and satel-lites at distance two from the center of each splitter.If the instance of 3-PARTITION has a solution, then the transmitter, col-lector and regions are drawn as illustrated in Fig. 13. The splitters of a 3-setwith a + a + a = B appear in one region and between the outer boundariesof the transmitter and the collector. Since each sector of the collector has B sectors, we can put a single satellite in each sector triangle and obtain a 1-planardrawing.Conversely, suppose there is a fan-crossing free drawing of G I , where thedrawing is on the sphere. As in [50], we place c T at the North Pole and c C atthe South Pole. A meridian path M i is a path of five fat edges between thecenters of the transmitter and the collector through the vertices u i and v Bi onthe boundaries, for i = 1 , . . . , m . Each meridian path also contains a unique pathof length five from c T to c C . A splitter path SP a = ( c T , a T , a (cid:48) T , c a , a (cid:48) C , a C , c C )is a path of length six between the centers of the transmitter and the collector22ia the center of a splitter P a , such that a T and a C are satellites of P a and a (cid:48) T and a (cid:48) C are vertices between a T and c a and c a and c C , respectively. Note thatthere is a splitter path from c C through a C and c a for a vertices and there is asingle path from c T through a T to c a .From now on assume that we are given a fan-crossing free embedding of G I .First, observe, that two meridian paths P and P (cid:48) do not cross, i,e., there is nocrossing of two edges e and e (cid:48) , where e is in the subgraph induced by P and e (cid:48) isin the subgraph induced by P (cid:48) . Edges e and e (cid:48) belong to two distinct K , and afan-crossing is unavoidable by the uniqueness of a fan-crossing free embeddingof K from Lemma 4 if P and P (cid:48) cross.Second, a meridian path P and a splitter path SP a do not cross. Towardsa contradiction, suppose that edge e of SP a crosses an edge f of some fat edgeof P . Each splitter path must cross the boundaries of the transmitter and thecollector, which needs a path of length at least four. If, in addition, an edge ofa K is crossed, then at least four more edge is needed. However, splitter pathsare tight and have length six.In consequence, we can follow the arguments given by Grigoriev and Bod-laender. Two successive meridian paths M i and M i +1 define a face, so thatthere is a cyclic ordering of faces according to the circuits of the boundaries ofthe transmitter and the collector. In consequence, there is a unique way to drawthe transmitter around the North Pole, and similarly, there is a unique way todraw the collector around the South Pole.We now consider the drawing of splitters. Each splitter path has length six.It takes a fan-crossing free path of length at least three from the center of thetransmitter to cross the the inner and outer circuit including the diagonal, andsimilarly for the collector. Hence, the center c a of a splitter must be placedbetween the outer circuits from c T and c C .In consequence, each satellite is placed in a sector triangle and each connectorin an outer triangle. Then there is at most one satellite in each sector triangle;otherwise there is a fan crossing, since the edges to the satellites are incident tothe centers c T and c C of the transmitter and collector, respectively.Hence, for each region, we have exactly three splitters, which each haveone satellite in a sector triangle, and there is at most one satellite in each ofthe B sector triangles of the region (face) of the collector. Since we have Bm such paths, each face must contain exactly three splitters with exactly B pathsbetween the splitter centers and the South Pole, which implies that the instance I has a solution. (cid:117)(cid:116) The recognition problem for fan-crossing free graphs with a fixed rotationsystem is also NP-complete. Here, a graph is given with a rotation systemdescribing the cyclic ordering of the edges incident to each vertex. Therefore,we can modify the proof of [16] for the NP-completeness of recognizing 1-planargraphs with a rotation system in a similar way as in the proof of Theorem 8.23 a) the general schema u i u ‘ i u ‘ i+1 u i+1 c T c a c T c a ‘ (b) a detailed viewFigure 13: Illustration of graph G I for the NP-reduction
5. Conclusion
In this work, we have shown that the fan-crossing free graphs are incom-parable with the fan-crossing, 2-planar, and 1-gap-planar graphs, respectively.There were not listed in the table of relationships in [40]. It remains to showthat there are fan-crossing free graphs that are not quasi-planar graphs. More-over, we have added fan-crossing free graphs to the list of beyond-planar graphclasses with an NP-complete recognition problem, which was open so far [40].We have shown a closure resp. non-closure of graph classes under subdivi-sion, node-to-circle expansion, and path-addition, respectively, and have intro-duced the notion of universality, which is used for a specification of beyond-planarity.Many new graph classes can be defined by the intersection of two (or more)graph classes or by combining the respective properties in a drawing. Such ”dou-ble” classes are largely unexplored, as remarked in [40]. Some facts are knownabout 1-planar and RAC graphs, where IC-planar combines both properties [33].However, there are graphs that are 1-planar and RAC and are not IC-planar, forexample, n -vertex tiles, which have too many edges for IC-planarity. There areNIC-planar that are 1-planar and not RAC [17]. On the other hand, n -vertexRAC graphs with 4 n −
10 edges [45] and n -vertex fan-crossing free graphs with4 n −
6. Acknowledgements
I wish to thank Therese Biedl for point out the simple box-visibility represen-tation, and to Michael Bekos for drawing my attention to the crossing numberof cube-connected cycles.
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