The constructive characterization of \((k,l)\)-edge-connected digraphs (Q653995)

A digraph is called \(k\)-edge-connected if deleting any \(k -1\) edges leaves it strongly connected. A digraph \(G = (V,E)\) is called \((k,l)\)-edge-connected for some integers \(0\leq l\leq k\), if \(G\) has a root vertex \(s\) and for each vertex \(z\neq s\), there exist \(k\) edge-disjoint \(sz\) paths and \(l\) edge-disjoint \(zs\) paths. The paper deals with a constructive characterization of \((k,l)\)-edge-connectedness. The main result provides two operations that allow to construct any graph belonging to the studied class. Moreover each graph obtained in terms of described construction is \((k,l)\)-edge-connected. It proves the conjecture of \textit{A.~Frank} and \textit{L. Szegő} [Discrete Appl. Math. 131, No. 2, 347--371 (2003; Zbl 1022.05071)].