# Decomposing rooted (k,l)-connected graphs into rooted k-connected parts

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Let *G=(V,E)* be an undirected graph, and [math]r \in V[/math] a root node. *G* is called rooted *(k,l)*-connected if *G-X* is [math](k-\vert X\vert)l[/math]-edge-connected for any [math]X \subseteq V-r[/math]. Is it true that every rooted *(k,l)*-connected graph contains *l* edge-disjoint spanning rooted *k*-connected subgraphs?

## Remarks

A similar statement for connectivity between two nodes was shown to be true by Egawa, Kaneko, and Matsumoto ^{[1]}. Similar connectivity properties also have been studied in ^{[2]} and ^{[3]}.

## References

- ↑ Y. Egawa, A. Kaneko, M. Matsumoto,
*A mixed version of Menger's theorem*, Combinatorica 11 (1991), 71-74. DOI link - ↑ A. Kaneko, K. Ota,
*On minimally (n, λ)-connected graphs*, J. Combin. Theory, Series B 80 (2000), 156-171. DOI link - ↑ A.R. Berg, T. Jordán,
*Sparse certificates and removable cycles in l-mixed p-connected graphs*, Oper. Res. Lett. 33 (2005), 111-114. DOI link EGRES Tech. Report