# Calculating Crossing Numbers of Graphs Using Their Redrawings

## Abstract

**:**

## 1. Introduction

## 2. Cyclic Permutations and Possible Drawings of ${\mathit{G}}^{*}$

**Lemma**

**1.**

**Proof.**

- Let us first suppose that $2s+t\le 2\lceil \frac{n}{2}\rceil $; that is, $-2s-t\ge -2\lceil \frac{n}{2}\rceil $. The number of crossings in D satisfies$${\mathrm{cr}}_{D}({G}^{*}+{D}_{n})={\mathrm{cr}}_{D}\left({K}_{6,n}\right)+{\mathrm{cr}}_{D}({K}_{6,n},{G}^{*})+{\mathrm{cr}}_{D}\left({G}^{*}\right)\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+1s+2t+3(n-s-t)$$$$=6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+3n-2s-t\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+3n-2\u2308\frac{n}{2}\u2309\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b.$$
- Now, let $2s+t>2\lceil \frac{n}{2}\rceil $, which yields that $2s+t\ge 2\lceil \frac{n}{2}\rceil +1$ and also that $s\ge 1$. By fixing the subgraph ${G}^{*}\cup {T}^{i}$ for some ${T}^{i}\in {S}_{D}$, we have$${\mathrm{cr}}_{D}({G}^{*}+{D}_{n})={\mathrm{cr}}_{D}\left({K}_{6,n-1}\right)+{\mathrm{cr}}_{D}({K}_{6,n-1},{G}^{*}\cup {T}^{i})+{\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i})$$$$\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+6(s-1)+5t+4(n-s-t)+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+4n+2s+t-5$$$$\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+4n+2\u2308\frac{n}{2}\u2309+1-5\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b,$$

**Corollary**

**1.**

**Corollary**

**2.**

## 3. The Crossing Number of ${\mathit{G}}^{*}+{\mathit{D}}_{\mathit{n}}$

**Lemma**

**2.**

**Theorem**

**1.**

**Proof.**

**Case 1:**${\mathrm{cr}}_{D}\left({G}^{*}\right)=1$. Let us first consider the subdrawing of ${G}^{*}$ induced by D given in Figure 4a. Since the set ${R}_{D}\cup {S}_{D}$ is nonempty, two possible subcases may occur.

- (a)
- Let ${R}_{D}$ be the nonempty set; that is, there is a subgraph ${T}^{i}\in {R}_{D}$. The reader can easily see that the subgraph ${F}^{i}={G}^{*}\cup {T}^{i}$ is uniquely represented by ${\mathrm{rot}}_{D}\left({t}_{i}\right)=\left(153462\right)$. By fixing the subgraph ${G}^{*}\cup {T}^{i}$, if edges of ${G}^{*}\cup {T}^{i}$ are crossed by any other subgraph ${T}^{j}$ at least five times, we obtain$${\mathrm{cr}}_{D}({G}^{*}+{D}_{n})={\mathrm{cr}}_{D}\left({K}_{5,n-1}\right)+{\mathrm{cr}}_{D}({K}_{5,n-1},{G}^{*}\cup {T}^{i})+{\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i})$$$$\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+5(n-1)+1\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b.$$If there is some subgraph ${T}^{j}$ with ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i},{T}^{j})<5$, then the vertex ${t}_{j}$ cannot be placed in the outer region of subdrawing $D\left({G}^{*}\right)$ with all six vertices of ${G}^{*}$ on its boundary, and ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i},{T}^{j})=4$ enforces ${\mathrm{cr}}_{D}({T}^{i},{T}^{j})=0$. Thus, by fixing the subgraph ${T}^{i}\cup {T}^{j}$, we have$${\mathrm{cr}}_{D}({G}^{*}+{D}_{n-2})+{\mathrm{cr}}_{D}({T}^{i}\cup {T}^{j})+{\mathrm{cr}}_{D}({K}_{6,n-2},{T}^{i}\cup {T}^{j})+{\mathrm{cr}}_{D}({G}^{*},{T}^{i}\cup {T}^{j})$$$$\ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+n-2+2\u230a\frac{n-2}{2}\u230b+6(n-2)+4=6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b,$$
- (b)
- Let ${R}_{D}$ be the empty set; that is, there is a subgraph ${T}^{i}\in {S}_{D}$. As $s\ge 1$, we deal with possible configurations ${\mathcal{A}}_{p}$ from the nonempty set ${\mathcal{M}}_{D}$. For any $p\in \{1,2,3,4\}$, if there is a subgraph ${T}^{j}$, $j\ne i$ such that ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i},{T}^{j})<5$ and ${\mathrm{cr}}_{D}({T}^{i},{T}^{j})=0$ with $\mathrm{conf}\left({F}^{i}\right)={\mathcal{A}}_{p}$, the same fixation of ${T}^{i}\cup {T}^{j}$ like in the previous case also confirms a contradiction with (3) in D.Now, let us turn to the possibility of obtaining the minimum value 4 in Table 1; that is, $\mathrm{cr}({\mathcal{A}}_{p},{\mathcal{A}}_{q})=4$ could be achieved in D for two different ${\mathcal{A}}_{p},{\mathcal{A}}_{q}\in {\mathcal{M}}_{D}$. In the rest of the paper, assume that there are two different subgraphs ${T}^{i},\phantom{\rule{0.277778em}{0ex}}{T}^{j}\in {S}_{D}$ such that ${F}^{i}$ and ${F}^{j}$ have mentioned configurations ${\mathcal{A}}_{p}$ and ${\mathcal{A}}_{q}$, respectively. Then, ${\mathrm{cr}}_{D}({T}^{i}\cup {T}^{j},{T}^{k})\ge 9$ holds for any ${T}^{k}\in {S}_{D}$ with $k\ne i,j$ by summing two corresponding values in Table 1. We can easily verify in six possible regions of $D({G}^{*}\cup {T}^{i})$ and $D({G}^{*}\cup {T}^{j})$ that ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i},{T}^{k})\ge 2+3=5$ and ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{j},{T}^{k})\ge 2+3=5$ are fulfilling for any ${T}^{k}\in {T}_{D}$, which yields that ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i}\cup {T}^{j},{T}^{k})\ge 2+3+3=8$ trivially holds for any such subgraph ${T}^{k}$. Moreover, each of $n-s-t$ subgraphs ${T}^{k}\notin {S}_{D}\cup {T}_{D}$ of ${K}_{6,n-2}$ crosses ${G}^{*}\cup {T}^{i}\cup {T}^{j}$ at least six times. As ${\mathrm{cr}}_{D}({G}^{*}\cup {T}^{i}\cup {T}^{j})\ge 7$, by fixing the subgraph ${G}^{*}\cup {T}^{i}\cup {T}^{j}$, we have$${\mathrm{cr}}_{D}({G}^{*}+{D}_{n})\ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+10(s-2)+8t+6(n-s-t)+7$$$$\ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+6n+2\left(2\u2308\frac{n}{2}\u2309+1\right)-13\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b,$$The obtained number of crossings contradicts the assumption (3). Finally, let us consider that $\mathrm{cr}({\mathcal{A}}_{p},{\mathcal{A}}_{q})\ge 5$ holds for all ${\mathcal{A}}_{p},{\mathcal{A}}_{q}\in {\mathcal{M}}_{D}$ with $p,q\in \{1,2,3,4\}$. By fixing the subgraph ${G}^{*}\cup {T}^{i}$ for some ${T}^{i}\in {S}_{D}$, we have$${\mathrm{cr}}_{D}({G}^{*}+{D}_{n})\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+6(s-1)+5t+4(n-s-t)+2$$$$\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+4n+2\u2308\frac{n}{2}\u2309+1-4\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b.$$This again confirms a contradiction with (3) in D.

**Case 2:**${\mathrm{cr}}_{D}\left({G}^{*}\right)\ge 2$. For all such subdrawings of the graph ${G}^{*}$ in Figure 4 and Figure 5, if all six vertices of ${G}^{*}$ are included in one region of $D\left({G}^{*}\right)$ and the set ${R}_{D}$ is nonempty, then the same technique like in the first part of Case 1 can be applied. To finish the proof of this case, let ${R}_{D}$ be the empty set. Let any subgraph ${T}^{i}\in {S}_{D}$ be crossed at least once by each other subgraph ${T}^{j}$, because otherwise fixing ${T}^{i}\cup {T}^{j}$ results in at least $6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b$ crossings in D. This assumption solves the problem of the drawing of ${G}^{*}$ given in Figure 4b described above after the proof of Lemma 1. Finally, for all remaining subdrawings of ${G}^{*}$ induced by D with any ${T}^{i}\in {S}_{D}$, we can verify over all possible regions of $D({G}^{*}\cup {T}^{i})$ that the edges of ${G}^{*}\cup {T}^{i}$ are crossed at least five times by each other subgraph ${T}^{j}$, $j\ne i$. Again, by fixing the subgraph ${G}^{*}\cup {T}^{i}$, we have

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Elimination of two crossings on edges of ${G}^{*}$ with vertex notation in a different order for both bottom drawings. (

**a**): the subdrawing of ${G}^{*}$ with two crossings on four edges ${v}_{1}{v}_{5},{v}_{3}{v}_{5},{v}_{2}{v}_{6},{v}_{4}{v}_{6}$; (

**b**): the subdrawing of ${G}^{*}$ with four crossings on four edges ${v}_{1}{v}_{5},{v}_{3}{v}_{5},{v}_{2}{v}_{6},{v}_{4}{v}_{6}$.

**Figure 2.**Elimination of two crossings on edges of ${G}^{*}$ with vertex notation in a different order for both bottom drawings, after which, edges of ${C}_{4}\left({G}^{*}\right)$ do not cross each other. (

**a**): elimination of one crossing on edges of ${C}_{4}\left({G}^{*}\right)$ with ${\mathrm{cr}}_{D}\left({G}^{*}\right)=7$; (

**b**): elimination of one crossing on edges of ${C}_{4}\left({G}^{*}\right)$ with ${\mathrm{cr}}_{D}\left({G}^{*}\right)=5$; (

**c**): elimination of two crossings on edges of ${C}_{4}\left({G}^{*}\right)$.

**Figure 3.**Elimination of two crossings on edges of ${G}^{*}$ with vertex notation in a different order for both bottom drawings, after which, edges of ${C}_{4}\left({G}^{*}\right)$ cross each other. (

**a**): elimination of two crossings on edges of ${C}_{4}\left({G}^{*}\right)$ with ${\mathrm{cr}}_{D}\left({G}^{*}\right)=5$; (

**b**): elimination two crossings on edges of ${C}_{4}\left({G}^{*}\right)$ with ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$.

**Figure 4.**Five considered nonplanar drawings of the graph ${G}^{*}$ in which edges of ${C}_{4}\left({G}^{*}\right)$ cross each other. (

**a**): the drawing of ${G}^{*}$ with all six vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=1$; (

**b**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=2$; (

**c**): the drawing of ${G}^{*}$ with all six vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$; (

**d**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=5$; (

**e**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$.

**Figure 5.**Seven considered nonplanar drawings of the graph ${G}^{*}$ in which edges of ${C}_{4}\left({G}^{*}\right)$ do not cross each other. (

**a**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=1$; (

**b**): the drawing of ${G}^{*}$ with all six vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$; (

**c**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$; (

**d**): the drawing of ${G}^{*}$ with all six vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$; (

**e**): the drawing of ${G}^{*}$ with all six vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$; (

**f**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$; (

**g**): the drawing of ${G}^{*}$ with five vertices of ${G}^{*}$ located in one region of $D\left({G}^{*}\right)$ and ${\mathrm{cr}}_{D}\left({G}^{*}\right)=3$.

**Figure 6.**Drawings of four possible configurations ${\mathcal{A}}_{p}$ of subgraph ${F}^{i}$ for ${T}^{i}\in {S}_{D}$.

**Figure 7.**Drawings of four possible configurations ${\mathcal{B}}_{p}$ of subgraph ${F}^{i}$ for ${T}^{i}\in {S}_{D}$.

**Figure 8.**The good drawing of ${G}^{*}+{D}_{n}$ with $6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b$ crossings for n odd.

**Figure 9.**The good drawing of ${G}^{*}+{D}_{n}$ with $6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+2\u230a\frac{n}{2}\u230b$ crossings for n even.

**Table 1.**The minimum number of crossings between ${T}^{i}$ and ${T}^{j}$ for two configurations ${\mathcal{X}}_{p}$ and ${\mathcal{X}}_{q}$ of subgraphs ${F}^{i}={G}^{*}\cup {T}^{i}$ and ${F}^{j}={G}^{*}\cup {T}^{j}$, where $\mathcal{X}=\mathcal{A}$ and $\mathcal{X}=\mathcal{B}$ for configurations in $\mathcal{M}$ and $\mathcal{N}$, respectively.

- | ${\mathcal{X}}_{1}$ | ${\mathcal{X}}_{2}$ | ${\mathcal{X}}_{3}$ | ${\mathcal{X}}_{4}$ |
---|---|---|---|---|

${\mathcal{X}}_{1}$ | 6 | 4 | 4 | 5 |

${\mathcal{X}}_{2}$ | 4 | 6 | 5 | 4 |

${\mathcal{X}}_{3}$ | 4 | 5 | 6 | 4 |

${\mathcal{X}}_{4}$ | 5 | 4 | 4 | 6 |

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**MDPI and ACS Style**

Staš, M.
Calculating Crossing Numbers of Graphs Using Their Redrawings. *Symmetry* **2023**, *15*, 175.
https://doi.org/10.3390/sym15010175

**AMA Style**

Staš M.
Calculating Crossing Numbers of Graphs Using Their Redrawings. *Symmetry*. 2023; 15(1):175.
https://doi.org/10.3390/sym15010175

**Chicago/Turabian Style**

Staš, Michal.
2023. "Calculating Crossing Numbers of Graphs Using Their Redrawings" *Symmetry* 15, no. 1: 175.
https://doi.org/10.3390/sym15010175