Transversal (combinatorics): Difference between revisions

Revision as of 05:50, 27 August 2022

Revision as of 04:06, 3 January 2023

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* One variation is that there is a [[bijection]] ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a '''system of distinct representatives''' (SDR).<ref name="lp">{{Cite Lovasz Plummer}}</ref>{{rp|29}}

* The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. In this situation, the members of the '''system of representatives''' are not necessarily distinct.<ref>{{citation|last1=Roberts|first1=Fred S.|title=Applied Combinatorics|year=2009|edition=2nd|place=Boca Raton|publisher=CRC Press|isbn=978-1-4200-9982-9|last2=Tesman|first2=Barry}}</ref>{{Rp|692}}<ref>{{citation|last=Brualdi|first=Richard A.|title=Introductory Combinatorics|year=2010|edition=5th|place=Upper Saddle River, NJ|publisher=Prentice Hall|isbn=978-0-13-602040-20}}</ref>{{Rp|322}}

In [[computer science]], computing transversals is useful in several application domains, with the input [[family of sets]] often being described as a [[hypergraph]].

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A '''partial transversal''' is a set containing at most one element from each member of the collection, or (in the stricter form of the concept) a set with an injection from the set to ''C''. The transversals of a finite collection ''C'' of finite sets form the basis sets of a [[matroid]], the '''transversal matroid''' of ''C''. The independent sets of the transversal matroid are the partial transversals of ''C''.<ref>{{citation|last=Oxley|first=James G.|title=Matroid Theory|volume=3|page=48|year=2006|series=Oxford graduate texts in mathematics|publisher=Oxford University Press|isbn=978-0-19-920250-8|authorlink=James Oxley}}.</ref>

An '''independent transversal''' (also called a '''[[rainbow-independent set]]''' or '''independent system of representatives''') is a transversal which is also an [[Independent set (graph theory)|independent set]] of a given graph. To explain the difference in figurative terms, consider a faculty with ''m'' departments, where the faculty dean wants to construct a committee of ''m'' members, one member per department. Such a committee is a transversal. But now, suppose that some faculty members dislike each other and do not agree to sit in the committee together. In this case, the committee must be an independent transversal, where the underlying graph describes the "dislike" relations.<ref>{{Cite journal|last=Haxell|first=P.|date=2011-11-01|title=On Forming Committees|url=https://www.tandfonline.com/doi/abs/10.4169/amer.math.monthly.118.09.777|journal=The American Mathematical Monthly|volume=118|issue=9|pages=777–788|doi=10.4169/amer.math.monthly.118.09.777|s2cid=27202372 |issn=0002-9890}}</ref>

Another generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of ''C''. An example of the latter would be a '''[[Bernstein set]]''', which is defined as a set that has a non-empty intersection with each set of ''C'', but contains no set of ''C'', where ''C'' is the collection of all [[perfect set]]s of a topological [[Polish space]]. As another example, let ''C'' consist of all the lines of a [[projective plane]], then a [[blocking set]] in this plane is a set of points which intersects each line but contains no line.