Skip to main content
SpringerLink
Log in

A rooted-forest partition with uniform vertex demand

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

A rooted-forest is the union of vertex-disjoint rooted-trees. Suppose we are given a graph G=(V,E), a collection {R 1,…,R k } of k root-sets (i.e., vertex-sets), and a positive integer d. We prove a necessary and sufficient condition for G to contain k edge-disjoint rooted-forests F 1,…,F k with root-sets R 1,…,R k such that each vertex is spanned by exactly d of F 1,…,F k .

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bérczi K, Frank A (2009) Packing arborescences. Technical Report 2009-04, EGRES

  • Berg A, Jordán T (2003) Algorithms for graph rigidity and scene analysis. In: Proceedings of the 11th annual European symposium on algorithms (ESA). Lecture notes in computer science, vol 2832. Springer, Berlin, pp 78–89

    Google Scholar 

  • Crapo H (1990) On the generic rigidity of plane frameworks. Technical report, Institut National de Recherche en Informatique et en Automatique

  • Edmonds J (1970) Submodular functions, matroids, and certain polyhedra. In: Guy R, Hanani H, Sauer N, Schönheim J (eds) Combinatorial structures and their applications. Gordon and Breach, New York, pp 69–87

    Google Scholar 

  • Edmonds J (1973) Edge disjoint branchings. In: Rustin B (ed) Combinatorial algorithms. Algorithmics Press, New York, pp 91–96

    Google Scholar 

  • Fekete Z, Szegö L (2004) A note on [k,l]-sparse graphs. In: Graph theory in Paris; a conference in memory of Claude Berge, pp 169–177

    Google Scholar 

  • Frank A, Szegö L (2003) Constructive characterizations for packing and covering with trees. Discrete Appl Math 131(2):347–371

    Article  MathSciNet  MATH  Google Scholar 

  • Fujishige S (2010) A note on disjoint arborescences. Combinatorica. doi:10.1007/s00493-010-2518-y

    MathSciNet  Google Scholar 

  • Gabow H, Tarjan R (1988) A linear-time algorithm for finding a minimum spanning pseudoforest. Inform Process Lett 27(5):259–263

    Article  MathSciNet  MATH  Google Scholar 

  • Gabow H, Westermann H (1992) Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica 7(1):465–497

    Article  MathSciNet  MATH  Google Scholar 

  • Haas R (2002) Characterizations of arboricity of graphs. Ars Combin 63:129–138

    MathSciNet  MATH  Google Scholar 

  • Hakimi S (1965) On the degrees of the vertices of a directed graph. J Franklin Inst 279(4):290–308

    Article  MathSciNet  MATH  Google Scholar 

  • Haller K, Lee A, Sitharam M, Streinu I, White N (2009) Body-and-cad geometric constraint systems. In: Proceedings of the 25th ACM symposium on applied computing. ACM, New York, pp 1127–1131

    Chapter  Google Scholar 

  • Imai H (1983) Network flow algorithms for lower truncated transversal polymatroids. J Oper Res Soc Japan 26(3):186–210

    MathSciNet  MATH  Google Scholar 

  • Jackson B, Jordán T (2009) Graph theoretic techniques in the analysis of uniquely localizable sensor networks. In: Mao G, Fidan B (eds) Localization algorithms and strategies for wireless sensor networks. IGI Global, Hershey, pp 146–173

    Chapter  Google Scholar 

  • Kamiyama N, Katoh N, Takizawa A (2009) Arc-disjoint in-trees in directed graphs. Combinatorica 29(2):197–214

    MathSciNet  MATH  Google Scholar 

  • Katoh N, Tanigawa S (2009) On the infinitesimal rigidity of bar-and-slider frameworks. In: Proceedings of the 20th international symposium on algorithms and computation (ISAAC 2009). Lecture notes in computer science, vol 5878. Springer, Berlin, pp 524–533

    Google Scholar 

  • Laman G (1970) On graphs and rigidity of plane skeletal structures. J Eng Math 4(4):331–340

    Article  MathSciNet  MATH  Google Scholar 

  • Lee A, Streinu I (2008) Pebble game algorithms and sparse graphs. Discrete Math 308(8):1425–1437

    Article  MathSciNet  MATH  Google Scholar 

  • Nash-Williams C (1961) Edge-disjoint spanning trees of finite graphs. J Lond Math Soc 1(1):445–450

    Article  MathSciNet  Google Scholar 

  • Oxley J (1992) Matroid Theory. Oxford University Press, London

    MATH  Google Scholar 

  • Pym J, Perfect H (1970) Submodular functions and independence structures. J Math Anal Appl 30(1–31):33

    MathSciNet  Google Scholar 

  • Recski A (1988) Network theory approach to the rigidity of skeletal structures. Part ii. Laman’s theorem and topological formulae. Discrete Appl Math 8(1):63–68

    Article  MathSciNet  Google Scholar 

  • Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin

    MATH  Google Scholar 

  • Streinu I, Theran L (2009) Sparsity-certifying graph decompositions. Graphs Combin 25(2):219–238

    Article  MathSciNet  MATH  Google Scholar 

  • Sugihara K (1985) Detection of structural inconsistency in systems of equations with degrees of freedom and its applications. Discrete Appl Math 10(3):297–312

    Article  MathSciNet  MATH  Google Scholar 

  • Tay T (1984) Rigidity of multi-graphs. I: Linking rigid bodies in n-space. J Combin Theory Ser B 36(1):95–112

    Article  MathSciNet  MATH  Google Scholar 

  • Tay T (1989) Linking (n−2)-dimensional panels in n-space II: (n−2,2)-frameworks and body and hinge structures. Graphs Combin 5(1):245–273

    Article  MathSciNet  MATH  Google Scholar 

  • Tay T (1993) A new proof of Laman’s theorem. Graphs Combin 9(2):365–370

    Article  MathSciNet  MATH  Google Scholar 

  • Tutte WT (1961) On the problem of decomposing a graph into n connected factors. J Lond Math Soc 36:221–230

    Article  MathSciNet  MATH  Google Scholar 

  • Whiteley W (1988) The union of matroids and the rigidity of frameworks. SIAM J Discrete Math 1(2):237–255

    Article  MathSciNet  MATH  Google Scholar 

  • Whiteley W (2005) Counting out to the flexibility of molecules. Phys Biol 2:S116–S126

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Architecture and Architectural Engineering, Kyoto University, Kyoto, Japan

    Naoki Katoh

  2. Research Institute for Mathematical Science, Kyoto University, Kyoto, Japan

    Shin-ichi Tanigawa

Authors
  1. Naoki Katoh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shin-ichi Tanigawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shin-ichi Tanigawa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katoh, N., Tanigawa, Si. A rooted-forest partition with uniform vertex demand. J Comb Optim 24, 67–98 (2012). https://doi.org/10.1007/s10878-010-9367-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-010-9367-x

Keywords

Search

Navigation