Journal of Combinatorial Optimization

, Volume 26, Issue 2, pp 372–384 | Cite as

Objective functions with redundant domains

  • Fatima Affif Chaouche
  • Carrie Rutherford
  • Robin Whitty


Let \((E,{ \mathcal{A}})\) be a set system consisting of a finite collection \({ \mathcal{A}}\) of subsets of a ground set E, and suppose that we have a function ϕ which maps \({ \mathcal{A}}\) into some set S. Now removing a subset K from E gives a restriction \({ \mathcal{A}}(\bar{K})\) to those sets of \({ \mathcal{A}}\) disjoint from K, and we have a corresponding restriction \(\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{K})}\) of our function ϕ. If the removal of K does not affect the image set of ϕ, that is \(\mbox {Im}(\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{X})})=\mbox {Im}(\phi)\), then we will say that K is a kernel set of \({ \mathcal{A}}\) with respect to ϕ. Such sets are potentially useful in optimisation problems defined in terms of ϕ. We will call the set of all subsets of E that are kernel sets with respect to ϕ a kernel system and denote it by \(\mathrm {Ker}_{\phi}({ \mathcal{A}})\). Motivated by the optimisation theme, we ask which kernel systems are matroids. For instance, if \({ \mathcal{A}}\) is the collection of forests in a graph G with coloured edges and ϕ counts how many edges of each colour occurs in a forest then \(\mathrm {Ker}_{\phi}({ \mathcal{A}})\) is isomorphic to the disjoint sum of the cocycle matroids of the differently coloured subgraphs; on the other hand, if \({ \mathcal{A}}\) is the power set of a set of positive integers, and ϕ is the function which takes the values 1 and 0 on subsets according to whether they are sum-free or not, then we show that \(\mathrm {Ker}_{\phi}({ \mathcal{A}})\) is essentially never a matroid.


Matroid Optimization Objective function Duality Turán-type problems 



The generalisation of Example 5 to give Proposition 6 was suggested to us by Tony Forbes. We are pleased to acknowledge helpful discussions with Bill Jackson and Taoyang Wu. The work of Chung and Graham on universal graphs was brought to our attention by Emil Vaughan and by Francesca Merola. An anonymous referee corrected a good many errors and made some helpful suggestions on presentation.


  1. Chung FRK, Graham RL (1979) On universal graphs. Ann NY Acad Sci 319:136–140 MathSciNetCrossRefGoogle Scholar
  2. Chung FRK, Graham RL (1983) On universal graphs for spanning trees. J Lond Math Soc 27(2):203–211 MathSciNetMATHCrossRefGoogle Scholar
  3. Chung FRK, Graham RL (1998) Erdős on graphs: his legacy of unsolved problems. AK Peters, Wellesley Google Scholar
  4. Frank A (1979) Kernel systems of directed graphs. Acta Sci Math 41:63–76 MATHGoogle Scholar
  5. Geelen J, Gerards AMH, Whittle G (2007) Towards a matroid-minor structure theory. In: Grimmett G et al. (eds) Combinatorics, complexity, and chance: a tribute to dominic welsh. Oxford lecture series in mathematics and its applications, vol 34. Oxford University Press, London, pp 72–82 CrossRefGoogle Scholar
  6. Green B (2004) The Cameron–Erdős conjecture. Bull Lond Math Soc 36:769–778 MATHCrossRefGoogle Scholar
  7. Oxley JG (1992) Matroid theory. Oxford University Press, London MATHGoogle Scholar
  8. Sapozhenko AA (2003) The Cameron–Erdős conjecture. Dokl Akad Nauk 393(6):749–752 MathSciNetGoogle Scholar
  9. Welsh DJA (1976) Matroid theory. Oxford University Press, London MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Fatima Affif Chaouche
    • 1
  • Carrie Rutherford
    • 2
  • Robin Whitty
    • 2
  1. 1.University of Sciences and Technology Houari BoumedieneAlgiersAlgeria
  2. 2.London South Bank UniversityLondonUK

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