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A new bound on maximum independent set and minimum connected dominating set in unit disk graphs

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Abstract

It is a conjecture that in any unit disk graph \(G\), \(\alpha (G) \le 3 \cdot \gamma _c(G) + 3\) where \(\alpha (G)\) is the size of the maximum independent set in \(G\) and \(\gamma _c(G)\) is the size of minimum connected dominating set in \(G\). In this paper, we show that in any unit disk graph \(G\), \(\alpha (G) \le 3.399 \cdot \gamma _c(G) + 4.874\). Currently, this is the best-known bound.

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References

  • Cadei M, Cheng MX, Cheng X, Du D-Z (2002) Connected domination in ad hoc wireless networks. In: Proceedings the Sixth International Conference on Computer Science and Informatics (CS &I’2002)

  • Clark BN, Colbourn CJ, Johnson DS (1990) Unit disk graphs. Discret Math 86:165–177

    Article  MATH  MathSciNet  Google Scholar 

  • Funke S, Kesselman A, Meyer U (2006) A simple improved distributed algorithm for minimum CDS in unit disk graphs. ACM Trans Sens Netw 2:444–453

    Article  Google Scholar 

  • Gao X, Wang Y, Li X, Wu W (2009) Analysis on theoretical bounds for approximating dominating set problems. Discret Math Algorithms Appl 1(1):71–84

    Article  MATH  MathSciNet  Google Scholar 

  • Li M, Wan P-J, Yao FF (2009) Tighter approximation bounds for minimum CDS in wireless ad hoc networks, ISAAC’2009. LNCS 5878:699–709

    Google Scholar 

  • Li YS, Thai MT, Wang F, Yi C-W, Wan P-J, Du D-Z (2005) On greedy construction of connected dominating sets in wireless networks. Wiley J Wirel Commun Mob Comput 5:927–932

    Article  Google Scholar 

  • Min M, Huang X, Huang C-H, Wu W, Du H, Jia X (2006) Improving construction for connected dominating set with Steiner tree in wireless sensor networks. J Glob Optim 35:111–119

    Article  MATH  MathSciNet  Google Scholar 

  • Vahdatpour A, Dabiri F, Moazeni M, Sarrafzadeh M (2008) Theoretical bound and practical analysis of connected dominating set in ad hoc and sensor networks, In Proceedings of 22nd International Symposium on Distributed Computing (DISC), pp. 481–495

  • Wan P-J, Alzoubi KM, Frieder O (2004) Distributed construction of connected dominating set in wireless ad hoc networks. ACM Springer Mob Netw Appl 9(2):141–149 A preliminary version of this paper appeared in IEEE INFOCOM, 2002

    Article  Google Scholar 

  • Wan P-J, Wang L, Yao F (2008) Two-phased approximation algorithms for minimum CDS in wireless ad hoc networks. IEEE ICDCS 2008:337–344

    Google Scholar 

  • Wu W, Du H, Jia X, Li Y, Huang SC-H (2006) Minimum connected dominating sets and maximal independent sets in unit disk graphs. Theor Comput Sci 352(1–3):1–7

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang Z, Wu W, Ding L, Liu Q, Wu L (2013) Packing circles in circles and applications, Handbook of Combinatorial Optimization, 2nd edn, pp. 2549–2584

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Acknowledgments

We wish to thank Dr. Yan Huang for her supervision on our research work and reviewers for their insightful comments.

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Authors and Affiliations

  1. Highland Park High School, Dallas, TX, USA

    Yingfan L. Du & Hongmin W. Du

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  1. Yingfan L. Du
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  2. Hongmin W. Du
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Correspondence to Yingfan L. Du.

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Du, Y.L., Du, H.W. A new bound on maximum independent set and minimum connected dominating set in unit disk graphs. J Comb Optim 30, 1173–1179 (2015). https://doi.org/10.1007/s10878-013-9690-0

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