## 1. Introduction

## 2. Basic Results

**Theorem 1.**

**Theorem 2.**

**Theorem 3.**

**Theorem 4.**

**Theorem 5.**

**Observation 1.**

**Theorem 6.**

**Theorem 7.**

**Theorem 8.**

**Theorem 9.**

**Theorem 10.**

**Theorem 11.**

**Theorem 12.**

**Theorem 13.**

**Theorem 14.**

**Theorem 15.**

**Theorem 16.**

## 3. The Average Lower 2-Domination Number of Wheels Related Graphs

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Theorem 17.**

**Proof.**

**Remark 1.**

**Remark 2.**

**Theorem 18.**

**Proof.**

**Case 1.**

**Case 2.**

**Case 3.**

**Theorem 19.**

**Proof.**

**Theorem 20.**

**Proof.**

**Theorem 21.**

**Proof.**

## 4. An Algorithm for Computing the Average Lower 2-Domination Number

**BEGIN**

**END.**

**Example 1.**

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Mishkovski, I.; Biey, M.; Kocarev, L. Vulnerability of complex Networks. Commun. Nonlinear Sci. Numer Simulat.
**2011**, 16, 341–349. [Google Scholar] [CrossRef] - Newport, K.T.; Varshney, P.K. Design of survivable communication networks under performance constraints. IEEE Trans. Reliab.
**1991**, 40, 433–440. [Google Scholar] [CrossRef] - Turaci, T.; Okten, M. Vulnerability of Mycielski Graphs via Residual Closeness. Ars Comb.
**2015**, 118, 419–427. [Google Scholar] - Turaci, T. On the Average Lower Bondage Number a Graph. RAIRO-Oper. Res.
**2015**, in press. [Google Scholar] [CrossRef] - Frank, H.; Frisch, I.T. Analysis and design of survivable Networks. IEEE Trans. Commun. Technol.
**1970**, 18, 501–519. [Google Scholar] [CrossRef] - Chvatal, V. Tough graphs and Hamiltonian circuits. Discrete Math.
**1973**, 5, 215–228. [Google Scholar] [CrossRef] - Barefoot, C.A.; Entringer, R.; Swart, H. Vulnerability in graphs-a comparative survey. J. Combin. Math. Combin. Comput.
**1987**, 1, 13–22. [Google Scholar] - Haynes, T.W.; Hedeniemi, S.T.; Slater, P.J. Fundamentals of Domination in Graphs; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Aytaç, A.; Turacı, T.; Odabaş, Z.N. On the Bondage Number of Middle Graphs. Math. Notes
**2013**, 93, 803–811. [Google Scholar] [CrossRef] - Aytaç, A.; Odabas, Z.N.; Turacı, T. The Bondage Number for Some Graphs. C. R. Lacad. Bulg. Sci.
**2011**, 64, 925–930. [Google Scholar] - Turaci, T.; Okten, M. The edge eccentric connectivity index of hexagonal cactus chains. J. Comput. Theor. Nanosci.
**2015**, 12, 3977–3980. [Google Scholar] [CrossRef] - Aytaç, A.; Turacı, T. Vertex Vulnerability Parameter of Gear Graphs. Int. J. Found. Comput. Sci.
**2011**, 22, 1187–1195. [Google Scholar] [CrossRef] - Henning, M.A. Trees with Equal Average Domination and Independent Domination Numbers. Ars Comb.
**2004**, 71, 305–318. [Google Scholar] - Aslan, E.; Kırlangıç, A. The Average Lower Domination Number of Graphs. Bull. Int. Math. Virtual Inst.
**2013**, 3, 155–160. [Google Scholar] - Aytaç, V. Average Lower Domination Number in Graphs. C. R. Lacad. Bulg. Sci.
**2012**, 65, 1665–1674. [Google Scholar] - Blidia, M.; Chellali, M.; Maffray, F. On Average Lower Independence and Domination Number in Graphs. Discrete Math.
**2005**, 295, 1–11. [Google Scholar] [CrossRef] - Tuncel, G.H.; Turaci, T.; Coskun, B. The Average Lower Domination Number and Some Results of Complementary Prisms and Graph Join. J. Adv. Res. Appl. Math.
**2015**, 7, 52–61. [Google Scholar] - Beineke, L.W.; Oellermann, O.R.; Pippert, R.E. The Average Connectivity of a Graph. Discrete Math.
**2002**, 252, 31–45. [Google Scholar] [CrossRef] - Aslan, E. The Average Lower Connectivity of Graphs. J. Appl. Math.
**2014**, 2014. [Google Scholar] [CrossRef] - Bauer, D.; Harary, F.; Nieminen, J.; Suffel, C.L. Domination alteration sets in graph. Discrete Math.
**1983**, 47, 153–161. [Google Scholar] [CrossRef] - Chellali, M. Bounds on the 2-Domination Number in Cactus Graps. Opusc. Math.
**2006**, 26, 5–12. [Google Scholar] - Fink, J.F.; Jacobson, M.S. n-Domination in Graphs. In Graph Theory with Applications to Algorithms and Computer Science; Alavi, Y., Schwenk, A.J., Eds.; Wiley: New York, NY, USA, 1984; pp. 283–300. [Google Scholar]
- Turaci, T. On the Average Lower 2-domination Number a Graph. 2015; submitted. [Google Scholar]
- Turaci, T. The Concept of Vulnerability in graphs and Average Lower 2-domination Number. In Proceedings of the 28th National Mathematics Conference, Antalya, Turkey, 7–9 September 2015.
- Yang, Z.; Liu, Y.; Li, X.Y. Beyond trilateration: On the localizability of wireless ad-hoc networks. In Proceedings of the IEEE INFOCOM 2009, Rio de Janeiro, Brazil, 19–25 April 2009.
- Aytaç, A.; Odabaş, Z.N. Residual Closeness of Wheels and Related Networks. Int. J. Found. Comput. Sci.
**2011**, 22, 1229–1240. [Google Scholar] [CrossRef] - Javaid, I.; Shokat, S. On the Partition Dimension of Some Wheel Related Graphs. J. Prime Res. Math.
**2008**, 4, 154–164. [Google Scholar] - Krzywkowski, M. 2-Bondage in graphs. Int. J. Comput. Math.
**2013**, 90, 1358–1365. [Google Scholar] [CrossRef] - Prather, R.E. Discrete Mathematical Structures for Computer Science; Houghton Mifflin: Boston, MA, USA, 1976. [Google Scholar]

**Figure 2.**Graphs ${W}_{4},\text{\hspace{0.05em}\hspace{0.05em}}{G}_{4},\text{\hspace{0.05em}\hspace{0.05em}}{f}_{4},\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}{H}_{4}$ and $S{f}_{4}$.

**Figure 3.**Values of ${\mathsf{\gamma}}_{2av}({G}_{n})$, ${\mathsf{\gamma}}_{2av}({f}_{n})$ and ${\mathsf{\gamma}}_{2av}({H}_{n})$.

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