# Mathematical Properties on the Hyperbolicity of Interval Graphs

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Previous Results

**Lemma**

**1.**

**Corollary**

**1.**

**Lemma**

**2.**

**Corollary**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

- $\delta \left(G\right)=0$ if and only if G is a tree.
- $\delta \left(G\right)=3/4$ if and only if G is not a tree and every cycle in G has length 3.

**Corollary**

**3.**

**Theorem**

**5.**

**Corollary**

**4.**

**Theorem**

**6.**

**Theorem**

**7.**

## 3. Interval Graphs and Hyperbolicity

**Lemma**

**3.**

**Corollary**

**5.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**6.**

## 4. Interval Graphs with Edges of Length 1

**Corollary**

**7.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**9.**

- $\delta \left(G\right)=0$ if and only if G has the 0-intersection property.
- $\delta \left(G\right)=3/4$ if and only if G has the $(3/4)$-intersection property.
- $\delta \left(G\right)=1$ if and only if G has the 1-intersection property.
- $\delta \left(G\right)=5/4$ if and only if G does not have the 0, $3/4$, 1 and $(3/2)$-intersection properties.
- $\delta \left(G\right)=3/2$ if and only if G has the $(3/2)$-intersection property.

#### Complement of Interval Graphs

**Lemma**

**4.**

**Proof.**

**Theorem**

**10.**

- If $\mathrm{diam}V\left(G\right)=1$, then $\delta (\phantom{\rule{0.166667em}{0ex}}\overline{G}\phantom{\rule{0.166667em}{0ex}})=0$.
- If $2\le \mathrm{diam}V\left(G\right)\le 3$, then $0\le \delta (\phantom{\rule{0.166667em}{0ex}}\overline{G}\phantom{\rule{0.166667em}{0ex}})\le 2$.
- If $\mathrm{diam}V\left(G\right)\ge 4$, then $5/4\le \delta (\phantom{\rule{0.166667em}{0ex}}\overline{G}\phantom{\rule{0.166667em}{0ex}})\le 3/2$.

**Proof.**

**Corollary**

**8.**

**Corollary**

**9.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Hernández-Gómez, J.C.; Reyes, R.; Rodríguez, J.M.; Sigarreta, J.M.
Mathematical Properties on the Hyperbolicity of Interval Graphs. *Symmetry* **2017**, *9*, 255.
https://doi.org/10.3390/sym9110255

**AMA Style**

Hernández-Gómez JC, Reyes R, Rodríguez JM, Sigarreta JM.
Mathematical Properties on the Hyperbolicity of Interval Graphs. *Symmetry*. 2017; 9(11):255.
https://doi.org/10.3390/sym9110255

**Chicago/Turabian Style**

Hernández-Gómez, Juan C., Rosalío Reyes, José M. Rodríguez, and José M. Sigarreta.
2017. "Mathematical Properties on the Hyperbolicity of Interval Graphs" *Symmetry* 9, no. 11: 255.
https://doi.org/10.3390/sym9110255