Combinatorica

, Volume 29, Issue 1, pp 27–48 | Cite as

The superstar packing problem

Original Paper
First Online:
Received:
Revised:

Abstract

Hell and Kirkpatrick proved that in an undirected graph, a maximum size packing by a set of non-singleton stars can be found in polynomial time if this star-set is of the form {S 1, S 2, ..., S k } for some k∈ℤ+ (S i is the star with i leaves), and it is NP-hard otherwise. This may raise the question whether it is possible to enlarge a set of stars not of the form {S 1, S 2, ..., S k } by other non-star graphs to get a polynomially solvable graph packing problem. This paper shows such families of depth 2 trees. We show two approaches to this problem, a polynomial alternating forest algorithm, which implies a Berge-Tutte type min-max theorem, and a reduction to the degree constrained subgraph problem of Lovász.

Mathematics Subject Classification (2000)

05C70 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Cornuéjols: General factors of graphs, J. Combin. Theory Ser. B 42 (1988), 285–296.Google Scholar
  2. [2]
    G. Cornuéjols and D. Hartvigsen: An extension of matching theory, J. Combin. Theory Ser. B 40 (1986), 285–296.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    G. Cornuéjols, D. Hartvigsen and W. Pulleyblank: Packing subgraphs in a graph, Oper. Res. Letter 1 (1981/82), 139–143.CrossRefGoogle Scholar
  4. [4]
    D. Hartvigsen, P. Hell and J. Szabó: The k-piece packing problem, J. Graph Theory 52 (2006), 267–293.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Hell and D. G. Kirkpatrick: On the complexity of general graph factor problems, SIAM J. Computing 12 (1983), 601–609.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    P. Hell and D. G. Kirkpatrick: Packings by complete bipartite graphs, SIAM J. Algebraic and Discrete Math. 7 (1986), 113–129.CrossRefMathSciNetGoogle Scholar
  7. [7]
    M. Janata and J. Szabó: Generalized star packing problems, EGRES Technical Reports TR-2004-17 (2004), http://www.cs.elte.hu/egres/.
  8. [8]
    M. Las Vergnas: An extension of Tutte’s 1-factor theorem, Discrete Math. 23 (1978), 241–255.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Loebl and S. Poljak: On matroids induced by packing subgraphs, J. Combin. Theory Ser. B 44 (1988), 338–354.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Loebl and S. Poljak: Efficient subgraph packing, J. Combin. Theory Ser. B 59 (1993), 106–121.Google Scholar
  11. [11]
    L. Lovász: The factorization of graphs, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), (1970), pp. 243–246.Google Scholar
  12. [12]
    L. Lovász: The factorization of graphs II, Acta Math. Acad. Sci. Hungar. 23 (1972), 223–246.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    J. Szabó: A note on the degree prescribed factor problem, EGRES Technical Reports TR-2004-19 (2004), http://www.cs.elte.hu/egres/.
  14. [14]
    J. Szabó: Graph packings and the degree prescribed subgraph problem, PhD thesis, Eötvös University, Budapest, 2006.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2009

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  2. 2.MTA-ELTE Egerváry Research Group (EGRES) Institute of MathematicsEötvös UniversityBudapestHungary

Personalised recommendations