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A similar definition can be given for hypergraphs. Let ''H=(V,E)'' be a hypergraph and ''k'' a positive integer. ''H'' is called '''k-partition-connected''' if for every partition <math>{\mathcal P}</math> of ''V'' there are at least <math>k(|{\mathcal P}|-1)</math> hyperedges in ''E'' that are not induced by a class of <math>{\mathcal P}</math>. A 1-partition-connected hypergraph is also called '''partition-connected''' (note that not all connected hypergraphs are partition-connected).
Frank, Király and Kriesell <ref>A. Frank, T. Király, M. Kriesell, ''On decomposing a hypergraph into k connected sub-hypergraphs'', Discrete Applied Mathematics 131 (2003), 373-383. [http:name="FrKiKr"//dx.doi.org/10.1016/S0166-218X(02)00463-8 DOI link]. [http://www.cs.elte.hu/egres/tr/egres-01-02.pdf EGRES Technical Report].</ref> proved the following characterization: A hypergraph is ''k''-partition-connected if and only if it can be decomposed into ''k'' partition-connected spanning sub-hypergraphs. In contrast, for any ''k'' there are ''k''-edge-connected hypergraphs that cannot be decomposed into 2 connected spanning sub-hypergraphs<ref>J. Bang-Jensen, S. Thomassé, ''Highly connected hypergraphs containing no two edge-disjoint spanning connected subhypergraphs'', Discrete Applied Mathematics 131 (2003), 555-559. [http:name="BaTh"//dx.doi.org/10.1016/S0166-218X(02)00582-6 DOI link]. </ref>.
===(k,l)-partition-connectivity===
==References==
<references />
<ref name="FrKiKr">A. Frank, T. Király, M. Kriesell, ''On decomposing a hypergraph into k connected sub-hypergraphs'', Discrete Applied Mathematics 131 (2003), 373-383. [http://dx.doi.org/10.1016/S0166-218X(02)00463-8 DOI link], [http://www.cs.elte.hu/egres/tr/egres-01-02.pdf EGRES Technical Report]</ref>
<ref name="BaTh">J. Bang-Jensen, S. Thomassé, ''Highly connected hypergraphs containing no two edge-disjoint spanning connected subhypergraphs'', Discrete Applied Mathematics 131 (2003), 555-559. [http://dx.doi.org/10.1016/S0166-218X(02)00582-6 DOI link] </ref>
</references>
[[Category:Definitions]]
