Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Algorithmic and structural aspects of the P 3-Radon number

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The generalization of classical results about convex sets in ℝn to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P 3-convexity on graphs. P 3-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radon’s classical convexity result. We establish hardness results and describe efficient algorithms for trees.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Barbosa, R. M., Coelho, E. M. M., Dourado, M. C., Rautenbach, D., & Szwarcfiter, J. L. (2012) On the Carathéodory number for the convexity of paths of order three. SIAM Jounal on Discrete Mathematics, 26, 929–939.

    Article  Google Scholar 

  • Centeno, C. C., Dourado, M. C., Penso, L. D., Rautenbach, D., & Szwarcfiter, J. L. (2011). Irreversible conversion of graphs. Theoretical Computer Science, 412, 3693–3700.

    Article  Google Scholar 

  • Chellali, M., Favaron, O., Hansberg, A., & Volkmann, L. (2012). k-Domination and k-independence in graphs: a survey. Graphs and Combinatorics, 28, 1–55. doi:10.1007/s00373-011-1040-3.

    Article  Google Scholar 

  • Cook, S. A. (1971). The complexity of theorem-proving procedures. In Proc. 3rd ann. ACM sympos. theory computing, Shaker Heights, Ohio, 1971 (pp. 151–158). New York: ACM.

    Chapter  Google Scholar 

  • Dreyer, P. A., & Roberts, F. S. (2009). Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics, 157, 1615–1627.

    Article  Google Scholar 

  • Duchet, P. (1988). Convex sets in graphs. II: Minimal path convexity. Journal of Combinatorial Theory. Series B, 44, 307–316.

    Article  Google Scholar 

  • Erdős, P., Fried, E., Hajnal, A., & Milner, E. C. (1972). Some remarks on simple tournaments. Algebra Universalis, 2, 238–245.

    Article  Google Scholar 

  • Farber, M., & Jamison, R. E. (1986). Convexity in graphs and hypergraphs. SIAM Journal on Algebraic and Discrete Methods, 7, 433–444.

    Article  Google Scholar 

  • Karp, R. M. (1972). Reducibility among combinatorial problems. In Complexity of computer computations (pp. 85–103). New York: Plenum.

    Chapter  Google Scholar 

  • Moon, J. W. (1972). Embedding tournaments in simple tournaments. Discrete Mathematics, 2, 389–395.

    Article  Google Scholar 

  • Parker, D. B., Westhoff, R. F., & Wolf, M. J. (2008). On two-path convexity in multipartite tournaments. European Journal of Combinatorics, 29, 641–651.

    Article  Google Scholar 

  • Radon, J. (1921). Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Mathematische Annalen, 83, 113–115.

    Article  Google Scholar 

  • Varlet, J. C. (1976). Convexity in tournaments. Bulletin de la Société Royale Des Sciences de Liège, 45, 570–586.

    Google Scholar 

  • van de Vel, M. L. J. (1993). Theory of convex structures. Amsterdam: North-Holland.

    Google Scholar 

Download references

Acknowledgements

This work has been done in the context of the CAPES-DAAD PROBRAL project “Cycles, Convexity, and Searching in Graphs”. The first and fifth authors have been supported by CNPq and FAPERJ. The third author has been supported by FAPERJ. The sixth author has been supported by CAPES.

Author information

Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Mitre C. Dourado, Jayme L. Szwarcfiter & Alexandre Toman

  2. Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany

    Dieter Rautenbach & Philipp M. Schäfer

  3. PESC, COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Vinícius Fernandes dos Santos & Jayme L. Szwarcfiter

  4. NCE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Jayme L. Szwarcfiter

Authors
  1. Mitre C. Dourado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dieter Rautenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vinícius Fernandes dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Philipp M. Schäfer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jayme L. Szwarcfiter
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Alexandre Toman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dieter Rautenbach.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dourado, M.C., Rautenbach, D., dos Santos, V.F. et al. Algorithmic and structural aspects of the P 3-Radon number. Ann Oper Res 206, 75–91 (2013). https://doi.org/10.1007/s10479-013-1320-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-013-1320-9

Keywords

Search

Navigation