Rooted connectivity

For every global connectivity notion (edge-connectivity, node-connectivity, element-connectivity, in undirected/directed graphs/hypergraphs) we can define a rooted counterpart: there is a specified root node r, and disjoint paths are required only between pairs of nodes of which one is r. Some examples:

• For undirected hypergraphs, rooted k-edge-connectivity is equivalent to k-edge-connectivity.
• Rooted k-connectivity is not equivalent to k-connectivity even in graphs. For example [math]\{ru,ru,rv,rv\}[/math] is rooted 2-connected but not 2-connected.
• In directed hypergraphs, rooted k-arc-connectivity means that there are k arc-disjoint directed paths from r to any other node. We may also speak about rooted (k,l)-arc-connectivity: there are k arc-disjoint directed paths from r to any other node, and there are l arc-disjoint directed paths from any other node to r.