# Simple Realizability of Complete Abstract Topological Graphs Simplified

## Abstract

An *abstract topological graph* (briefly an *AT-graph*) is a pair \(A=(G,\mathcal {X})\) where \(G=(V,E)\) is a graph and \(\mathcal {X}\subseteq \left( {\begin{array}{c}E\\ 2\end{array}}\right) \) is a set of pairs of its edges. The AT-graph *A* is *simply realizable* if *G* can be drawn in the plane so that each pair of edges from \(\mathcal {X}\) crosses exactly once and no other pair crosses. We characterize simply realizable complete AT-graphs by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author.

## 1 Introduction

A *topological graph* \(T=(V(T), E(T))\) is a drawing of a graph *G* in the plane such that the vertices of *G* are represented by a set *V*(*T*) of distinct points and the edges of *G* are represented by a set *E*(*T*) of simple curves connecting the corresponding pairs of points. We call the elements of *V*(*T*) and *E*(*T*) the *vertices* and the *edges* of *T*, respectively. The drawing has to satisfy the following general position conditions: (1) the edges pass through no vertices except their endpoints, (2) every pair of edges has only a finite number of intersection points, (3) every intersection point of two edges is either a common endpoint or a proper crossing (“touching” of the edges is not allowed), and (4) no three edges pass through the same crossing. A topological graph or a drawing is *simple* if every pair of edges has at most one common point, which is either a common endpoint or a crossing. Simple topological graphs appear naturally as crossing-minimal drawings; it is well known that if two edges in a topological graph have more than one common point, then a local redrawing decreases the total number of crossings. A topological graph is *complete* if it is a drawing of a complete graph.

An *abstract topological graph* (briefly an *AT-graph*), a notion introduced by Kratochvíl, Lubiw and Nešetřil [10], is a pair \((G,\mathcal {X})\) where *G* is a graph and \(\mathcal {X} \subseteq \left( {\begin{array}{c}E(G)\\ 2\end{array}}\right) \) is a set of pairs of its edges. Here we assume that \(\mathcal {X}\) consists only of independent (that is, nonadjacent) pairs of edges. For a simple topological graph *T* that is a drawing of *G*, let \(\mathcal {X}_T\) be the set of pairs of edges having a common crossing. A simple topological graph *T* is a *simple realization* of \((G,\mathcal {X})\) if \(\mathcal {X}_T=\mathcal {X}\). We say that \((G,\mathcal {X})\) is *simply realizable* if \((G,\mathcal {X})\) has a simple realization.

An *AT-subgraph* of an AT-graph \((G,\mathcal {X})\) is an AT-graph \((H,\mathcal {Y})\) such that *H* is a subgraph of *G* and \(\mathcal {Y}=\mathcal {X}\cap \left( {\begin{array}{c}E(H)\\ 2\end{array}}\right) \). Clearly, a simple realization of \((G,\mathcal {X})\) restricted to the vertices and edges of *H* is a simple realization of \((H,\mathcal {Y})\).

We are ready to state our main result.

### **Theorem 1**

Every complete AT-graph that is not simply realizable has an AT-subgraph on at most six vertices that is not simply realizable.

We also show that AT-subgraphs with five vertices are not sufficient to characterize simple realizability.

### **Theorem 2**

There is a complete AT-graph *A* with six vertices such that all its induced AT-subgraphs with five vertices are simply realizable, but *A* itself is not.

Theorem 1 implies a straightforward polynomial algorithm for simple realizability of complete AT-graphs, running in time \(O(n^{6})\) for graphs with *n* vertices. It is likely that this running time can be improved relatively easily. However, compared to the first polynomial algorithm for simple realizability of complete AT-graphs [13], the new algorithm may be more suitable for implementation and for practical applications, such as generating all simply realizable complete *AT*-graphs of given size or computing the crossing number of the complete graph [5, 15]. On the other hand, the new algorithm does not directly provide the drawing itself, unlike the original algorithm [13]. The explicit list of realizable AT-graphs on six vertices can be generated using the database of small simple complete topological graphs created by Ábrego et al. [1].

For general noncomplete graphs, no such finite characterization by forbidden *AT*-subgraphs is possible. Indeed, in the special case when \(\mathcal {X}\) is empty, the problem of simple realizability is equivalent to planarity, and there are nonplanar graphs of arbitrarily large girth, such as subdivisions of \(K_5\). Moreover, simple realizability for general *AT*-graphs is NP-complete [11]. See [13] for an overview of other similar realizability problems.

The proof of Theorem 1 is based on the polynomial algorithm for simple realizability of complete AT-graphs from [13]. The main idea is very simple: every time the algorithm rejects the input, it is due to an obstruction of constant size.

Theorem 1 is an analogue of a similar characterization of simple monotone drawings of \(K_n\) by forbidden 5-tuples, and pseudo linear drawings of \(K_n\) by forbidden 4-tuples [4].

Ábrego et al. [1, 2] independently verified that simple complete topological graphs with up to nine vertices can be characterized by forbidden rotation systems of five-vertex subgraphs; see Sect. 2 for the definition. They conjectured that the same characterization is true for all simple complete topological graphs [2]. This conjecture now follows by combining their result for six-vertex graphs with Theorem 1. This gives a finite characterization of *realizable abstract rotation systems* defined in [14, Sect. 3.5], where it was also stated that such a characterization was not likely [14, p. 739]. The fact that only 5-tuples are sufficient for the characterization by rotation systems should perhaps not be too surprising, as rotation systems characterize simple drawings of \(K_n\) more economically, using only \(O(n^2\log n)\) bits, whereas AT-graphs need \(\varTheta (n^4)\) bits.

## 2 Preliminaries

Topological graphs *G* and *H* are *weakly isomorphic* if they are realizations of the same abstract topological graph.

The *rotation* of a vertex *v* in a topological graph is the clockwise cyclic order in which the edges incident with *v* leave the vertex *v*. The *rotation system* of a topological graph is the set of rotations of all its vertices. Similarly we define the *rotation* of a crossing *x* of edges *uv* and *yz* as the clockwise order in which the four parts *xu*, *xv*, *xy* and *xz* of the edges *uv* and *yz* leave the point *x*. Note that each crossing has exactly two possible rotations. We will represent the rotation of a vertex *v* as an ordered sequence of the endpoints of the edges incident with *v*. The *extended rotation system* of a topological graph is the set of rotations of all its vertices and crossings.

Assuming that *T* and \(T'\) are drawings of the same abstract graph, we say that their rotation systems are *inverse* if for each vertex \(v \in V(T)\), the rotation of *v* and the rotation of the corresponding vertex \(v' \in V(T')\) are inverse cyclic permutations. If *T* and \(T'\) are weakly isomorphic simple topological graphs, we say that their extended rotation systems are *inverse* if their rotation systems are inverse and, in addition, for every crossing *x* in *T*, the rotation of *x* and the rotation of the corresponding crossing \(x'\) in \(T'\) are inverse cyclic permutations. For example, if \(T'\) is a mirror image of *T*, then *T* and \(T'\) have inverse extended rotation systems.

We say that two cyclic permutations of sets *A*, *B* are *compatible* if they are restrictions of a common cyclic permutation of \(A\cup B\).

Simple complete topological graphs have the following key property.

### **Proposition 3**

**.**

- (1)
If two simple complete topological graphs are weakly isomorphic, then their extended rotation systems are either the same or inverse.

- (2)
For every edge

*e*of a simple complete topological graph*T*and for every pair of edges \(f,f'\in E(T)\) that have a common endpoint and cross*e*, the AT-graph of*T*determines the order of crossings of*e*with the edges \(f,f'\).

By inspecting simple drawings of \(K_4\), it can be shown that the converse of Proposition 3 also holds: the rotation system of a simple complete topological graph determines which pairs of edges cross [12, 16].

## 3 Proof of Theorem 2

*ij*to denote the edge \(\{i,j\}\). Let \(A=((V,E),\mathcal {X})\) be the complete AT-graph with vertex set \(V=\{0,1,2,3,4,5\}\) and with

*A*with five vertices is simply realizable; see Fig. 1.

*A*is not simply realizable. Suppose that

*T*is a simple realization of

*A*. Without loss of generality, assume that the rotation of 5 in \(T[\{1,2,3,5\}]\) is (1, 2, 3). By Proposition 3 and by the first drawing in Fig. 1, the rotation of 5 in \(T[\{1,2,3,4,5\}]\) is (1, 2, 3, 4), since the inverse would not be compatible with (1, 2, 3). Similarly, by the second drawing in Fig. 1 the rotation of 5 in \(T[\{0,2,3,4,5\}]\) is (2, 3, 0, 4), since the inverse would not be compatible with (1, 2, 3, 4). By the third drawing in Fig. 1, the rotation of 5 in \(T[\{0,1,3,4,5\}]\) is (0, 1, 3, 4) or (0, 4, 3, 1), but neither of them is compatible with both (1, 2, 3, 4) and (2, 3, 0, 4); a contradiction.

## 4 Proof of Theorem 1

Let \(A=(K_n,\mathcal {X})\) be a given complete abstract topological graph with vertex set \([n]=\{1,2,\dots ,n\}\). The algorithm from [13] for deciding simple realizability of *A* has the following three main steps: computing the rotation system, determining the homotopy class of every edge with respect to the edges incident with one chosen vertex *v*, and computing the number of crossings of every pair of edges in a crossing-optimal drawing with the rotation system and homotopy class fixed from the previous steps. We follow the algorithm and analyze each step in detail.

**Step 1: Computing the Extended Rotation System**

This step is based on the proof of Proposition 3; see [13, Proposition 3].

**1(a) Realizability of 5-tuples.** For every 5-tuple *Q* of vertices of *A*, the algorithm tests whether *A*[*Q*] is simply realizable. If not, then the 5-tuple certifies that *A* is not simply realizable. If *A*[*Q*] is simply realizable, then by Proposition 3, the algorithm computes a rotation system \(\mathcal {R}(Q)\) such that the rotation system of every simple realization of *A*[*Q*] is either \(\mathcal {R}(Q)\) or the inverse of \(\mathcal {R}(Q)\).

**1(b) Orienting 5-tuples.** For every 5-tuple \(Q\subseteq [n]\), the algorithm selects an orientation \(\varPhi (\mathcal {R}(Q))\) of \(\mathcal {R}(Q)\) so that for every pair of 5-tuples \(Q,Q'\) having four common vertices and for each \(x\in Q\cap Q'\), the rotations of *x* in \(\varPhi (\mathcal {R}(Q))\) and \(\varPhi (\mathcal {R}(Q'))\) are compatible. If there is no such orientation map \(\varPhi \), the AT-graph *A* is not simply realizable. We show that in this case there is a set *S* of six vertices of *A* that certifies this.

Let \(Q_1,Q_2\) be two 5-tuples with four common elements, let \(\mathcal {R}_1\) be a rotation system on \(Q_1\) and let \(\mathcal {R}_2\) be a rotation system on \(Q_2\). We say that \(\mathcal {R}_1\) and \(\mathcal {R}_2\) are *compatible* if for every \(x\in Q_1\cap Q_2\), the rotations of *x* in \(\mathcal {R}_1\) and \(\mathcal {R}_2\) are compatible.

Let \(\mathcal {G}\) be the graph with vertex set \(\left( {\begin{array}{c}[n]\\ 5\end{array}}\right) \) and edge set consisting of those pairs \(\{Q,Q'\}\) whose intersection has size 4. For every edge \(\{Q,Q'\}\) of \(\mathcal {G}\), at most one orientation of \(\mathcal {R}(Q')\) is compatible with \(\mathcal {R}(Q)\). If no orientation of \(\mathcal {R}(Q')\) is compatible with \(\mathcal {R}(Q)\), then the 6-tuple \(S=Q\cup Q'\) certifies that *A* is not simply realizable. We may thus assume that for every edge \(\{Q,Q'\}\) of \(\mathcal {G}\), exactly one orientation of \(\mathcal {R}(Q')\) is compatible with \(\mathcal {R}(Q)\). Let \(\mathcal {E}\) be the set of those edges \(\{Q,Q'\}\) of \(\mathcal {G}\) such that \(\mathcal {R}(Q)\) and \(\mathcal {R}(Q')\) are not compatible.

Call a set \(\mathcal {W} \subseteq \left( {\begin{array}{c}[n]\\ 5\end{array}}\right) \) *orientable* if there is an orientation map \(\varPhi \) assigning to every rotation system \(\mathcal {R}(Q)\) with \(Q\in \mathcal {W}\) either \(\mathcal {R}(Q)\) itself or its inverse \((\mathcal {R}(Q))^{-1}\), such that for every pair of 5-tuples \(Q,Q' \in \mathcal {W}\) with \(|Q\cap Q'|=4\), the rotation systems \(\varPhi (\mathcal {R}(Q))\) and \(\varPhi (\mathcal {R}(Q'))\) are compatible.

### **Lemma 4**

If \(\left( {\begin{array}{c}[n]\\ 5\end{array}}\right) \) is not orientable, then there is a 6-tuple \(S \subseteq [n]\) such that *S* 5 is not orientable.

### *Proof*

Clearly, \(\left( {\begin{array}{c}[n]\\ 5\end{array}}\right) \) is not orientable if and only if \(\mathcal {G}\) has a cycle with an odd number of edges from \(\mathcal {E}\). Call such a cycle a *nonorientable* cycle. We claim that if \(\mathcal {G}\) has a nonorientable cycle, then \(\mathcal {G}\) has a nonorientable triangle. Let \(\mathcal {C}(\mathcal {G})\) be the cycle space of \(\mathcal {G}\). The parity of the number of edges of \(\mathcal {E}\) in \(\mathcal {K}\in \mathcal {C}(\mathcal {G})\) is a linear form on \(\mathcal {C}(\mathcal {G})\). Hence, to prove our claim, it is sufficient to show that \(\mathcal {C}(\mathcal {G})\) is generated by triangles.

Suppose that \(\mathcal {K}=F_1F_2\dots F_k\), with \(k\ge 4\), is a shortest cycle in \(\mathcal {G}\) that is not a sum of triangles in \(\mathcal {C}(\mathcal {G})\). Then \(\mathcal {K}\) is an induced cycle in \(\mathcal {G}\), that is, \(|F_i \cap F_j|\le 3\) if \(2 \le |i-j| \le k-2\). Let \(z\in F_1 \setminus F_2\). Then \(z\in F_k\), otherwise \(|F_k\cap F_1\cap F_2|=4\). Let *i* be the smallest index such that \(i\ge 3\) and \(z\in F_i\). We have \(i\ge 4\), otherwise \(|F_1\cap F_3|=|(F_1\cap F_2\cap F_3)\cup \{z\}|=4\). For every \(j\in \{2,\dots ,i-2\}\), let \(F'_j=(F_j\cap F_{j+1}) \cup \{z\}\). Then \(\mathcal {K}\) is the sum of the closed walk \(\mathcal {K}'=F_1F'_2\dots F'_{i-2}F_i \dots F_k\) and the triangles \(F_1F_2F'_2,F_{i-1}F_iF'_{i-2}\), \(F_jF_{j+1}F'_j\) for \(j=2, \dots , i-2\) and \(F_{j+1}F'_jF'_{j+1}\) for \(j=2,\dots , i-3\); see Fig. 2. Since the length of \(\mathcal {K}'\) is \(k-1\), we have a contradiction with the choice of \(\mathcal {K}'\).

Let \(Q_1Q_2Q_3\) be a nonorientable triangle in \(\mathcal {G}\). The 5-tuples \(Q_1,Q_2,Q_3\) have either three or four common elements. Suppose that \(|Q_1\cap Q_2\cap Q_3|=4\) and let \(\{u,v,w,z\}=Q_1\cap Q_2\cap Q_3\). Then we may orient the rotation systems \(\mathcal {R}(Q_1), \mathcal {R}(Q_2)\) and \(\mathcal {R}(Q_3)\) so that the rotation of *u* in each of the orientations is compatible with (*v*, *w*, *z*). This implies that the rotations of *u* in the resulting rotation systems are pairwise compatible. Thus, the resulting rotation systems are pairwise compatible, a contradiction. Hence, we have \(|Q_1\cap Q_2\cap Q_3|=3\), which implies that \(|Q_1\cup Q_2\cup Q_3|=6\). Setting \(S=Q_1\cup Q_2\cup Q_3\), the set \(\left( {\begin{array}{c}S\\ 5\end{array}}\right) \) is not orientable.\(\square \)

If \(\left( {\begin{array}{c}[n]\\ 5\end{array}}\right) \) is orientable, there are exactly two possible solutions for the orientation map. We will assume that the rotation of 1 in \(\varPhi (\mathcal {R}(\{1,2,3,4,5\}))\) is compatible with (2, 3, 4), so that there is at most one solution \(\varPhi \).

**1(c) Computing the Rotations of Vertices.** Having oriented the rotation system of every 5-tuple, the algorithm now computes the rotation of every \(x\in [n]\), as the cyclic permutation compatible with the rotation of *x* in every \(\varPhi (\mathcal {R}(Q))\) such that \(x\in Q \in \left( {\begin{array}{c}[n]\\ 5\end{array}}\right) \). We show that this is always possible. The following lemma forms the core of the argument.

### **Lemma 5**

Let \(k\ge 4\). For every \(F\in \left( {\begin{array}{c}[k+1]\\ k\end{array}}\right) \), let \(\pi _F\) be a cyclic permutation of *F* such that for every pair \(F,F'\in \left( {\begin{array}{c}[k+1]\\ k\end{array}}\right) \), the cyclic permutations \(\pi _F\) and \(\pi _F'\) are compatible. Then there is a cyclic permutation \(\pi _{[k+1]}\) of \([k+1]\) compatible with all the cyclic permutations \(\pi _F\) with \(F\in \left( {\begin{array}{c}[k+1]\\ k\end{array}}\right) \).

**1(d) Computing the Rotations of Crossings.** For every pair of edges \(\{\{u,v\},\{x,y\}\}\in \mathcal {X}\), the algorithm determines the rotation of their crossing from the rotations of the vertices *u*, *v*, *x*, *y*. This finishes the computation of the extended rotation system.

**Step 2: Determining the Homotopy Classes of the Edges**

Let *v* be a fixed vertex of *A* and let *S*(*v*) be a topological star consisting of *v* and all the edges incident with *v*, drawn in the plane so that the rotation of *v* agrees with the rotation computed in the previous step. For every edge \(e=xy\) of *A* not incident with *v*, the algorithm computes the order of crossings of *e* with the subset \(E_{v,e}\) of edges of *S*(*v*) that *e* has to cross. By Proposition 3 (2), the five-vertex AT-subgraphs of *A* determine the relative order of crossings of *e* with every pair of edges of \(E_{v,e}\). Define a binary relation \(\prec _{x,y}\) on \(E_{v,e}\) so that \(vu \prec _{x,y} vw\) if the crossing of *e* with *vu* is closer to *x* than the crossing of *e* with *vw*. If \(\prec _{x,y}\) is acyclic, it defines a total order of crossings of *e* with the edges of \(E_{v,e}\). If \(\prec _{x,y}\) has a cycle, then it also has an oriented triangle \(vu_1,vu_2,vu_3\). This means that the AT-subgraph of *A* induced by the six vertices \(v,u_1,u_2,u_3,x,y\) is not simply realizable.

We recall that the *homotopy class* of a curve \(\varphi \) in a surface \(\varSigma \) relative to the boundary of \(\varSigma \) is the set of all curves that can be obtained from \(\varphi \) by a continuous deformation within \(\varSigma \), keeping the boundary of \(\varSigma \) fixed.

The *homotopy class of e* is determined by the following combinatorial data: the set \(E_{v,e}\), the total order \(\prec _{x,y}\) in which the edges of \(E_{v,e}\) cross *e*, the rotations of these crossings, and the rotations of the vertices *x* and *y*. Consider the star *S*(*v*) drawn on the sphere. Cut circular holes around the points representing all the vertices except *v*, and let \(\varSigma \) be the resulting surface with boundary. Let \(x_e\) and \(y_e\) be fixed points on the boundaries of the two holes around *x* and *y*, respectively, so that the orders of these points corresponding to all the edges of *A* on the boundaries of the holes agree with the computed rotation system. Draw a curve \(\varphi _e\) with endpoints \(x_e\) and \(y_e\) satisfying all the combinatorial data of *e*. Now the homotopy class of *e* is defined as the homotopy class of \(\varphi _e\) in \(\varSigma \) relative to the boundary of \(\varSigma \).

**Step 3: Computing the Minimum Crossing Numbers**

For every pair of edges *e*, *f*, let \(\mathrm{cr}(e,f)\) be the minimum possible number of crossings of two curves from the homotopy classes of *e* and *f*. Similarly, let \(\mathrm{cr}(e)\) be the minimum possible number of self-crossings of a curve from the homotopy class of *e*. The numbers \(\mathrm{cr}(e,f)\) and \(\mathrm{cr}(e)\) can be computed in polynomial time in any 2-dimensional surface with boundary [3, 17]. In our special case, the algorithm is relatively straightforward [13].

We use the key fact that from the homotopy class of every edge, it is possible to choose a representative such that the crossing numbers \(\mathrm{cr}(e,f)\) and \(\mathrm{cr}(e)\) are all realized simultaneously [13]. This is a consequence of the following facts.

### **Lemma 6**

[9]**.** Let \(\gamma \) be a curve on an orientable surface *S* with endpoints on the boundary of *S* that has more self-intersections than required by its homotopy class. Then there is a singular 1-gon or a singular 2-gon bounded by parts of \(\gamma \).

Here a *singular 1-gon* of a curve \(\gamma :[0,1] \rightarrow S\) is an image \(\gamma [\alpha ]\) of an interval \(\alpha \subset [0,1]\) such that \(\gamma \) identifies the endpoints of \(\alpha \) and the resulting loop is contractible in *S*. A *singular 2-gon* of \(\gamma \) is an image of two disjoint intervals \(\alpha , \beta \subset [0,1]\) such that \(\gamma \) identifies the endpoints of \(\alpha \) with the endpoints of \(\beta \) and the resulting loop is contractible in *S*.

### **Lemma 7**

[6, 9]**.** Let \(C_1\) and \(C_2\) be two simple curves on a surface *S* such that the endpoints of \(C_1\) and \(C_2\) lie on the boundary of *S*. If \(C_1\) and \(C_2\) have more intersections than required by their homotopy classes, then there is an innermost embedded 2-gon between \(C_1\) and \(C_2\), that is, two subarcs of \(C_1\) and \(C_2\) bounding a disc in *S* whose interior is disjoint with \(C_1\) and \(C_2\).

Whenever there is a singular 1-gon, a singular 2-gon, or an embedded innermost 2-gon in a system of curves on *S*, it is possible to eliminate the 1-gon or 2-gon by a homotopy of the corresponding curves, which decreases the total number of crossings.

For the rest of the proof, we fix a drawing *D* of *A* such that its rotation system is the same as the rotation system computed in Step 1, the edges of *S*(*v*) do not cross each other, every other edge is drawn as a curve in its homotopy class computed in Step 2, and under these conditions, the total number of crossings is the minimum possible. Then every edge *f* of *S*(*v*) crosses every other edge *e* at most once, and this happens exactly if \(\{e,f\}\in \mathcal {X}\). Moreover, for every pair of edges \(e_1, e_2\) not incident with *v*, the corresponding curves in *D* cross exactly \(\mathrm{cr}(e_1,e_2)\) times, and the curve representing \(e_1\) has \(\mathrm{cr}(e_1)\) self-crossings. Hence, *A* is simply realizable if and only if all the edges \(e_1,e_2\) not incident with *v* satisfy \(\mathrm{cr}(e_1)=0\), \(\mathrm{cr}(e_1,e_2)\le 1\), and \(\mathrm{cr}(e_1,e_2)=1 \Leftrightarrow \{e_1,e_2\}\in \mathcal {X}\). Moreover, if *A* is simply realizable, then *D* is a simple realization of *A*.

We further proceed in four substeps. Due to space limitations, we only include a short sketch of the substeps 3(b)–3(d).

**3(a) Characterization of the Homotopy Classes.** Let \(w_1, w_2, \dots , w_{n-1}\) be the vertices of *A* adjacent to *v* so that the rotation of *v* is \((w_1,w_2, \dots , w_{n-1})\). Let \(w_aw_b\) be an edge such that \(1\le a<b\le n-1\). Since every AT-subgraph of *A* with 4 or 5 vertices is simply realizable, we have the following conditions on the homotopy class of \(w_aw_b\). Refer to Fig. 3.

### **Observation 8**

Suppose that \(\{w_aw_b,vw_c\}\in \mathcal {X}\); that is, \(vw_c\in E_{v,w_aw_b}\). If \(a<c<b\), then the rotation of the crossing of \(w_aw_b\) with \(vw_c\) is \((w_a,w_c,w_b,v)\). If \(c<a\) or \(b<c\), then the rotation of the crossing is \((w_b,w_c,w_a,v)\). \(\square \)

Observation 8 implies that the homotopy class of the edge \(w_aw_b\) is determined by a permutation of \(E_{v,w_aw_b}\) that determines the order in which \(w_aw_b\) crosses the edges in \(E_{v,w_aw_b}\). The next observation further restricts this permutation.

### **Observation 9**

Suppose that \(vw_c,vw_d\in E_{v,w_aw_b}\). If \(a<c<d<b\), then \(vw_c \prec _{w_a,w_b} vw_d\). If (*c*, *d*, *a*, *b*) is compatible with \((1,2,\dots ,n-1)\) as cyclic permutations, then \(vw_d \prec _{w_a,w_b} vw_c\). \(\square \)

On the other hand, it is easy to see that every homotopy class satisfying Observations 8 and 9 has a representative that is a simple curve. Therefore, \(cr(w_aw_b)=0\).

**3(b) The Parity of the Crossing Numbers.** It can be shown that if \(e_1\) and \(e_2\) are independent edges not incident with *v*, then \(\mathrm{cr}(e_1,e_2)\) is odd if and only if \(\{e_1,e_2\}\in \mathcal {X}\). It can also be shown that if \(e_1\) and \(e_2\) are adjacent edges not incident with *v*, then \(\mathrm{cr}(e_1,e_2)\) is even. It follows that *A* is realizable if and only if every pair of edges in *D* crosses at most once.

**3(c) Multiple Crossings of Adjacent Edges.** Next we show that adjacent edges do not cross in *D*, otherwise some AT-subgraph of *A* with five vertices is not simply realizable. This part is rather straightforward, although the full proof is not short. Let \(w_aw_b\) and \(w_aw_c\) be two adjacent edges. By symmetry, we may assume that \(a<b<c\). We will consider cyclic intervals (*a*, *b*), (*b*, *c*) and \((c,a)=(c,n-1] \cup [1,a)\). We define the following subsets of \(E_{v,w_aw_b}\) and \(E_{v,w_aw_c}\). For each of the three cyclic intervals (*i*, *j*), let \(F_b(i,j)=\{vw_k\in E_{v,w_aw_b}; k\in (i,j)\}\) and \(F_c(i,j)=\{vw_k\in E_{v,w_aw_c}; k\in (i,j)\}\). We will also write \(\prec _b\) as a shortcut for \(\prec _{w_aw_b}\) and \(\prec _c\) as a shortcut for \(\prec _{w_aw_c}\). By symmetry, we have two general cases: (I) \(w_aw_b\) does not cross \(vw_c\) and \(w_aw_c\) does not cross \(vw_b\), and (II) \(w_aw_b\) does not cross \(vw_c\) and \(w_aw_c\) crosses \(vw_b\).

For case (I), one can observe the following conditions; we omit the proofs.

### **Observation 10**

**.**

- (1)
We have \(F_b(c,a) \subseteq F_c(c,a)\) and \(F_c(a,b) \subseteq F_b(a,b)\).

- (2)
The sets \(F_b(b,c)\) and \(F_c(b,c)\) are disjoint.

- (3)
If \(vw_d\in F_b(b,c)\) and \(vw_e\in F_c(b,c)\), then \(d<e\).

- (4)
Let \(vw_d\in F_b(a,b)\cap F_c(a,b)\) and \(vw_e\in F_b(c,a)\cap F_c(c,a)\). Then \(vw_d \prec _b vw_e \Leftrightarrow vw_d \prec _c vw_e\).

- (5)
Let \(vw_d\in F_c(a,b)\) and \(vw_e\in F_b(b,c)\). Then \(vw_d \prec _b vw_e\). Similarly, if \(vw_d\in F_b(c,a)\) and \(vw_e\in F_c(b,c)\), then \(vw_d \prec _c vw_e\).

We show that Observation 10 implies that \(\mathrm{cr}(w_aw_b, w_aw_c)=0\). Refer to Fig. 4. Start with drawing the edges \(w_aw_b\) and \(w_aw_c\) without crossing. Conditions (2) and (4) imply that there is a total order \(\prec \) on \(E_{v,w_aw_b} \cup E_{v,w_aw_c}\) that is a common extension of \(\prec _b\) and \(\prec _c\). Let \(vw_i\) be the \(\prec \)-largest element of \(F_b(c,a) \cup F_c(a,b)\). Condition (5) implies that all edges \(vw_j\) from \(F_b(b,c) \cup F_c(b,c)\) satisfy \(vw_i \prec vw_j\). Condition (1) implies that we can draw the edges \(vw_j\) with \(vw_j \preceq vw_i\) like in the figure. Conditions (2), (3) and (5) imply that we can draw the edges \(vw_j\) with \(vw_i \prec vw_j\) like in the figure. The remaining edges of *S*(*v*) can be drawn easily. In this way we obtain a simple drawing with noncrossing representatives of the homotopy classes of \(w_aw_b\) and \(w_aw_c\).

The analysis of case (II) is similar.

**3(d) Detecting Multiple Crossings of Independent Edges.** Finally, we show by induction that if two independent edges cross more than once, then there is a five-vertex AT-subgraph that forces this, possibly for a different pair of edges. In this part, we strongly rely on the established fact that adjacent edges do not cross in *D*. We avoid a tedious case analysis by not continuing in the approach chosen for adjacent pairs of edges.

Let \(e=w_aw_b\) and \(f=w_cw_d\) be two independent edges that cross more than once in *D*. In the subgraph of *D* formed by the two edges *e* and *f*, the vertices \(w_a,w_b,w_c,w_d\) are incident to a common face, since adjacent edges do not cross in *D* and every pair of the four vertices \(w_a,w_b,w_c,w_d\) is connected by an edge. We assume without loss of generality that \(w_a,w_b,w_c,w_d\) are incident to the outer face. That is, we may draw a simple closed curve \(\gamma \) containing the vertices \(w_a,w_b,w_c,w_d\) but no interior points of *e* or *f*, such that the relative interiors of *e* and *f* are inside \(\gamma \).

*e*and

*f*cross an even number of times. The edge

*e*splits the region inside \(\gamma \) into two regions, \(R_0(e)\) and \(R_1(e)\), where \(R_0(e)\) is the region that does not contain the endpoints of

*f*on its boundary. Similarly,

*f*splits the region inside \(\gamma \) into regions \(R_0(f)\) and \(R_1(f)\) where \(R_0(f)\) is the region that does not contain the endpoints of

*e*on its boundary.

By Lemma 7, there is an innermost embedded 2-gon between *e* and *f*. For brevity, we call an innermost embedded 2-gon shortly a *bigon*. For a bigon *B*, by \(B^o\) we denote the open region inside *B* and we call it the *inside of B*. There are four possible types of bigons between *e* and *f*, according to the regions \(R_i(e)\) and \(R_j(f)\) in which their insides are contained. For \(i,j\in \{0,1\}\), we call a bigon *B* an *ij-bigon* if \(B^o\subseteq R_i(e) \cap R_j(f)\); see Fig. 5.

Since *D* is a drawing realizing the crossing number \(\mathrm{cr}(e,f)\), there is at least one vertex of *D* inside every bigon. The graph induced by *v*, the endpoints of *e* and *f*, and a set of vertices intersecting all bigons, certifies that *e* and *f* have at least \(\mathrm{cr}(e,f)\) crossings forced by their homotopy classes.

The following lemma quickly solves the case when there is at least one 00-bigon between *e* and *f*.

### **Lemma 11**

If *e* and *f* cross evenly and there is a 00-bigon *B* between *e* and *f* in *D*, then there is a vertex \(w_i\) inside *B*, and the AT-subgraph of *A* induced by the 5-tuple \(Q=\{w_a,w_b,w_c,w_d,w_i\}\) is not simply realizable.

We are left with the case that there is no 00-bigon between *e* and *f*.

### **Observation 12**

If *e* and *f* cross evenly and at least twice in *D*, and there is no 00-bigon between *e* and *f*, then there is a 01-bigon and a 10-bigon between *e* and *f*.

For a subset *W* of vertices of *A* containing *v* and the endpoints of two edges *g* and *f*, let \(\mathrm{cr}_W(g,f)\) be the minimum possible number of crossings of two curves from the homotopy classes of *g* and *f* determined by *A*[*W*], by a procedure analogous to the one in Step 2.

The following lemma proves the induction step in the case when *e* and *f* cross an even number of times.

### **Lemma 13**

If *e* and *f* cross evenly and at least twice in *D*, and there is no 00-bigon between *e* and *f*, then there is a proper subset *W* of vertices of *A* and an edge *g* independent from *f* such that \(\mathrm{cr}_W(g,f) \ge 2\).

If *e* and *f* cross an odd number of times, one can easily find another pair of independent edges crossing evenly and more than once. We omit the details.

## Notes

### Acknowledgements

I thank Martin Balko for his comments on an earlier version of the manuscript. I also thank all the reviewers for their suggestions for improving the presentation.

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