# Operations on Oriented Maps

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Maps and Oriented Maps

**Proposition**

**1.**

**Example**

**1.**

- an oriented map: ${\mathcal{M}}_{o}:\phantom{\rule{0.166667em}{0ex}}=\{\mathrm{\Sigma},\langle r,R\rangle \}$, or
- as an (orientable) map: $\mathcal{M}=\{\mathrm{\Omega},\langle {r}_{0},{r}_{1},{r}_{2}\rangle \}$.

**Lemma**

**1**(Fundamental Lemma of Symmetries of Maps)

**Lemma**

**2**(Fundamental Lemma of Symmetries of Oriented Maps)

**Definition**

**1.**

**Definition**

**2.**

**Corollary**

**1.**

- The cardinality of each orbit of $\mathrm{Aut}\mathcal{M}$ on Ω is equal to the order of $\mathrm{Aut}\mathcal{M}$.
- $\left|\mathrm{Aut}\phantom{\rule{0.1}{0ex}}\mathcal{M}\right|$ is a divisor of $\left|\mathrm{\Omega}\right|$.
- The projection $\mathcal{M}\to \mathcal{M}/\mathrm{Aut}\mathcal{M}$ is a regular covering projection.
- The quotient $T\left(\mathcal{M}\right)=\mathcal{M}/\mathrm{Aut}\mathcal{M}=\{{\mathrm{\Omega}}^{\prime},\langle {r}_{0}^{\prime},{r}_{1}^{\prime},{r}_{2}^{\prime}\rangle \}$ is a degenerate flag graph, called the symmetry type graph.
- The order $k=|{\mathrm{\Omega}}^{\prime}|$ of $T\left(\mathcal{M}\right)$ is equal to $k=\left|\mathrm{\Omega}\right|/\left|\mathrm{Aut}\mathcal{M}\right|$, and $\mathcal{M}$ is a k-orbit map.

**Corollary**

**2.**

- The cardinality of each orbit of $\mathrm{Aut}{\mathcal{M}}_{o}$ on Σ is equal to the order of $\mathrm{Aut}{\mathcal{M}}_{o}$.
- $|\mathrm{Aut}{\mathcal{M}}_{o}|$ is a divisor of $\left|\mathrm{\Sigma}\right|$.
- The projection ${\mathcal{M}}_{o}\to {\mathcal{M}}_{o}/\mathrm{Aut}{\mathcal{M}}_{o}$ is a regular covering projection.
- The quotient ${T}_{o}\left({\mathcal{M}}_{o}\right)={\mathcal{M}}_{o}/\mathrm{Aut}{\mathcal{M}}_{o}=\{{\mathrm{\Sigma}}^{\prime},\langle {R}^{\prime},{r}^{\prime}\rangle \}$ is a degenerate arc graph, called the oriented symmetry type graph.
- The order ${k}_{o}=\left|{\mathrm{\Sigma}}^{\prime}\right|$ of ${T}_{o}\left({\mathcal{M}}_{o}\right)$ is equal to ${k}_{o}=\left|\mathrm{\Sigma}\right|/|\mathrm{Aut}{\mathcal{M}}_{o}|$, and ${\mathcal{M}}_{o}$ is a ${k}_{o}$-orbit oriented map.

**Example**

**2.**

## 2. Operations On Oriented Maps

#### 2.1. Orientation Reversal Re or *

#### 2.2. Dual Du and Improper Dual IDu

#### 2.3. Truncation ($\mathtt{Tr}$)

#### 2.4. Medial ($\mathtt{Me}$)

#### 2.5. Snub ($\mathtt{Sn}$)

#### 2.6. Chamfer $\mathtt{Ch}$

#### 2.7. One-Dimensional Subdivision $\mathtt{Su}\mathtt{1}$

#### 2.8. Composite Operations

## 3. Some Properties of the Operations and k-Orbit Oriented Maps

**Theorem**

**1.**

**Definition**

**3.**

**Problem**

**1.**

**Definition**

**4.**

**Problem**

**2.**

## 4. Edge-Transitive Oriented Maps

**Theorem**

**2.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A fragment of an oriented map. The arcs are shown as black vertices; the green arrows correspond to the action of R on the arcs, and blue edges, to the action of r. The dashed lines and white vertices are from the underlying unoriented map. Orientation is counterclockwise, indicated with arrows.

**Figure 2.**The cube may be considered as an oriented map determined by R and r, or as a map determined by ${r}_{0},{r}_{1},{r}_{2}$.

**Figure 3.**An example of a symmetry type graph of a regular map and an oriented symmetry type graph of an oriented regular map; in this case, they may be viewed as the symmetry/oriented symmetry type graph of the cube.

**Figure 4.**The four-sided pyramid represented as a map $\mathcal{M}$ and as an oriented map ${\mathcal{M}}_{o}$, with the corresponding symmetry type graph $T\left(\mathcal{M}\right)$ and oriented symmetry type graph ${T}_{o}\left({\mathcal{M}}_{o}\right)$.

**Figure 5.**Improper dual ($\mathtt{IDu}$) and dual ($\mathtt{Du}$) of an oriented map, local figure. In the center is a local portion of an oriented map, to the left is is the local picture of $\mathtt{IDu}$, and to the right is $\mathtt{Du}$. Because there is only one copy of $\mathrm{\Sigma}$ used in the construction of $\mathtt{IDu}$ and $\mathtt{Du}$, we have suppressed the 0 subscript on the arcs for clarity.

**Figure 13.**The 17 oriented symmetry type graphs on at most four vertices. Eight of these marked by (**) or (*) correspond to edge-transitive maps. The three marked by (**) correspond to the strong edge-transitive maps.

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**MDPI and ACS Style**

Pisanski, T.; Williams, G.; Berman, L.W.
Operations on Oriented Maps. *Symmetry* **2017**, *9*, 274.
https://doi.org/10.3390/sym9110274

**AMA Style**

Pisanski T, Williams G, Berman LW.
Operations on Oriented Maps. *Symmetry*. 2017; 9(11):274.
https://doi.org/10.3390/sym9110274

**Chicago/Turabian Style**

Pisanski, Tomaž, Gordon Williams, and Leah Wrenn Berman.
2017. "Operations on Oriented Maps" *Symmetry* 9, no. 11: 274.
https://doi.org/10.3390/sym9110274