Blockers for Noncrossing Spanning Trees in Complete Geometric Graphs

  • Chaya Keller
  • Micha A. Perles
  • Eduardo Rivera-Campo
  • Virginia Urrutia-Galicia


In this chapter, we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs and, in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: Being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.


Span Tree General Position Boundary Edge Relative Interior Boundary Vertex 
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  1. 1.
    F. Harary, A.J. Schwenk, Trees with Hamiltonian square. Mathematika 18, 138–140 (1971)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    F. Harary, A.J. Schwenk, The number of caterpillars. Disc. Math. 6, 359–365 (1973)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. Carmen Hernando, Complejidad de Estructuras Geom\(\mathrm{\acute{e}}\)tricas y Combinatorias, Ph.D. Thesis, Universitat Polit\(\mathrm{\acute{e}}\)ctnica de Catalunya, 1999 [in Spanish]. Available online at:
  4. 4.
    G. K\(\mathrm{\acute{a}}\)rolyi, J. Pach, G. T\(\mathrm{\acute{o}}\)th, Ramsey-type results for geometric graphs I. Discrete Comput. Geom. 18, 247–255 (1997)Google Scholar
  5. 5.
    C. Keller, M.A. Perles, On the smallest sets blocking simple perfect matchings in a convex geometric graph. Israel J. Math. 187, 465–484 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    C. Keller, M.A. Perles, Characterization of co-blockers for simple perfect matchings in a convex geometric graph, submitted. Available online at:
  7. 7.
    J.J. Montellano-Ballesteros, E. Rivera-Campo, On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids, to appear in Graphs and Combinatorics. DOI: 10.1007/s00373-012-1190-yGoogle Scholar
  8. 8.
    J. Pach, in Geometric Graph Theory, ed. by J.E. Goodman, J. O’Rourke. Handbook of Discrete and Computational Geometry, 2nd edn.  Chapter 10 (CRC Press, Boca Raton, FL, 2004), pp. 219–238

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Chaya Keller
    • 1
  • Micha A. Perles
    • 1
  • Eduardo Rivera-Campo
    • 2
  • Virginia Urrutia-Galicia
    • 2
  1. 1.Einstein Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IztapalapaMéxicoMexico

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