# General Properties on Differential Sets of a Graph

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Some Bounds for the Cardinality of $\mathit{S}$

**Lemma**

**1.**

**Proposition**

**1.**

- (a)
- S is a maximum differential set if and only if it is a minimum dominating set.
- (b)
- If S is a minimum dominating set, then $P{N}_{S}\left(v\right)\ne \varnothing $ for every $v\in S$.
- (c)
- If S is a minimum dominating set and $v\in S$ satisfies that $|P{N}_{S}\left(v\right)|=1$, then ${\mathrm{deg}}_{S}\left(v\right)=0$.

**Proof.**

**Lemma**

**2**

**.**Let $G=(V,E)$ be a graph. If S is a minimum (respectively, maximum) β-differential set of G, then $\left|B\left(S\right)\right|\ge \left(\lfloor \beta \rfloor +1\right)\left|S\right|$ (respectively, $\left|B\right(S\left)\right|\ge \lceil \beta \rceil \left|S\right|$).

**Proposition**

**2.**

- (1)
- If S is a minimum β-differential set, then $\left|S\right|\le {\displaystyle \frac{n}{\lfloor \beta \rfloor +2}}.$
- (2)
- If S is a maximum β-differential set, then $\left|S\right|\le {\displaystyle \frac{n}{\lceil \beta \rceil +1}}.$

**Proof.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

## 3. Relationships between $\mathbf{\beta}$, $\mathit{B}\left(\mathit{S}\right)$ and $\mathit{C}\left(\mathit{S}\right)$

**Lemma**

**3.**

- (1)
- for all $v\in B\left(S\right)$, ${\delta}_{C\left(S\right)}\left(v\right)\le \lfloor \beta \rfloor +1$.
- (2)
- for all $v\in C\left(S\right)$, ${\delta}_{C\left(S\right)}\left(v\right)\le \lfloor \beta \rfloor $.

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

- (a)
- If $D\left(G\right)=2$, then $n-\delta (1+\beta )\le {\partial}_{\beta}\left(G\right)\le \delta (\Delta -1-\beta )+1$.
- (b)
- If $D\left(G\right)=3$, then $2(\delta -\beta )\le {\partial}_{\beta}\left(G\right)\le (\Delta -\beta )(\Delta (\Delta -1)+1)$.

**Proof.**

**Proposition**

**9.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Proposition**

**10.**

- (a)
- $\Delta \left(G\right)=n-1$ if and only if ${\partial}_{\beta}\left(G\right)=n-(1+\beta )$.
- (b)
- $\Delta \left(G\right)=n-2$ if and only if ${\partial}_{\beta}\left(G\right)=n-(2+\beta )$.
- (c)
- If $\beta >1$, then $\Delta \left(G\right)=n-3$, if and only if ${\partial}_{\beta}\left(G\right)=n-(3+\beta )$.

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

- (a)
- $(\delta +1)(min\left\{{w}_{i}\right\})-1\le \delta \left({G}^{\prime}\right)\le \Delta \left({G}^{\prime}\right)\le (\Delta +1)(max\left\{{w}_{i}\right\})-1$.
- (b)
- $n(min\left\{{w}_{i}\right\})\le n\left({G}^{\prime}\right)={\sum}_{i=1}^{n}{w}_{i}\le n(max\left\{{w}_{i}\right\})$.

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Conclusions

- Let G be a graph of order n and minimum degree $\delta $, and $\beta \in (0,\delta )$. Then$$n\le (\beta +2){\partial}_{\beta}\left(G\right)+\beta (\beta +1){\partial}_{\beta -1}\left(G\right).$$
- Let G be a graph with order n. For any $k\ge 1$ and $\beta >k$, we have$$\Delta =n-k-2\iff {\partial}_{\beta}\left(G\right)=n-(2+k+\beta ).$$
- Let $G=(V,W,E)$ be a weighted graph and let ${G}^{\prime}$ be its associated simple graph. Then, for every $\alpha >0$, ${\partial}_{\alpha -1}^{W}\left(G\right)={\partial}_{\alpha -1}\left({G}^{\prime}\right)$. Moreover, if $S\subseteq V$ such that ${\partial}_{\alpha -1}^{W}\left(S\right)={\partial}_{\alpha -1}^{W}\left(G\right)$, there exists ${S}^{\prime}\subseteq {V}^{\prime}$ such that ${\partial}_{\alpha -1}\left({S}^{\prime}\right)={\partial}_{\alpha -1}\left({G}^{\prime}\right)$ and $|{S}^{\prime}|=|S|$.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bermudo, S.; Fernau, H. Computing the differential of a graph: Hardness, approximability and exact algorithms. Discret. Appl. Math.
**2014**, 165, 69–82. [Google Scholar] [CrossRef] - Mashburn, J.L.; Haynes, T.W.; Hedetniemi, S.M.; Hedetniemi, S.T.; Slater, P.J. Differentials in graphs. Util. Math.
**2006**, 69, 43–54. [Google Scholar] - Abreu-Blaya, R.; Bermudo, S.; Rodríguez, J.M.; Tourís, E. Topological Indices and f-Polynomials on Some Graph Products. Symmetry
**2021**, 13, 292. [Google Scholar] [CrossRef] - Basilio, L.A.; Bermudo, S.; Sigarreta, J.M. Bounds on the differential of a graph. Util. Math.
**2017**, 103, 319–334. [Google Scholar] - Bermudo, S.; De la Torre, L.; Martín-Caraballo, A.M.; Sigarreta, J.M. The differential of the strong product graphs. Int. J. Comput. Math.
**2015**, 92, 1124–1134. [Google Scholar] [CrossRef] - Bermudo, S.; Fernau, H. Lower bound on the differential of a graph. Discret. Math.
**2012**, 312, 3236–3250. [Google Scholar] [CrossRef][Green Version] - Bermudo, S.; Fernau, H. Combinatorics for smaller kernels: The differential of a graph. Theor. Comput. Sci.
**2015**, 562, 330–345. [Google Scholar] [CrossRef] - Bermudo, S.; Rodríguez, J.M.; Sigarreta, J.M. On the differential in graphs. Util. Math.
**2015**, 97, 257–270. [Google Scholar] - Pushpam, P.R.L.; Yokesh, D. Differential in certain classes of graphs. Tamkang J. Math.
**2010**, 41, 129–138. [Google Scholar] [CrossRef] - Sigarreta, J.M. Differential in Cartesian Product Graphs. Ars Comb.
**2016**, 126, 259–267. [Google Scholar] - Goddard, W.; Henning, M.A. Generalised domination and independence in graphs. Congr. Numer.
**1997**, 123, 161–172. [Google Scholar] - Zhang, C.Q. Finding critical independent sets and critical vertex subsets are polynomial problems. SIAM J. Discret. Math.
**1990**, 3, 431–438. [Google Scholar] [CrossRef][Green Version] - Slater, P.J. Enclaveless sets and MK-systems. J. Res. Natl. Bur. Stand.
**1977**, 82, 197–202. [Google Scholar] [CrossRef] [PubMed] - Haynes, T.W.; Hedetniemi, S.; Slater, P.J. Fundamentals of Domination in Graphs; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Haynes, T.W.; Hedetniemi, S.; Slater, P.J. Domination in Graphs: Advanced Topics; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Basilio, L.A.; Bermudo, S.; Leanõs, J.; Sigarreta, J.M. β-Differential of a Graph. Symmetry
**2017**, 9, 205. [Google Scholar] [CrossRef][Green Version] - Kaneko, A.; Kelmans, A.; Nishimura, T. On packing 3-vertex paths in a graph. J. Graph Theory
**2001**, 36, 175–197. [Google Scholar] [CrossRef]

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Basilio, L.A.; Bermudo, S.; Hernández-Gómez, J.C.; Sigarreta, J.M. General Properties on Differential Sets of a Graph. *Axioms* **2021**, *10*, 265.
https://doi.org/10.3390/axioms10040265

**AMA Style**

Basilio LA, Bermudo S, Hernández-Gómez JC, Sigarreta JM. General Properties on Differential Sets of a Graph. *Axioms*. 2021; 10(4):265.
https://doi.org/10.3390/axioms10040265

**Chicago/Turabian Style**

Basilio, Ludwin A., Sergio Bermudo, Juan C. Hernández-Gómez, and José M. Sigarreta. 2021. "General Properties on Differential Sets of a Graph" *Axioms* 10, no. 4: 265.
https://doi.org/10.3390/axioms10040265