We attempt to generalize a theorem of Nash-Williams stating that graph has k-arc-connected orientation if and only it is 2k-edge-connected. In strongly connected digraph we call an arc deletable its deletion leaves digraph. Given 3-edge-connected G, define Frank number f(G) be the minimum k such there exist orientations G with property every edge becomes in at least one these orientations. are ...

A linear-time certifying algorithm for 3-edge-connectivity is presented. Given a connected undirected graph G, if G 3-edge-connected, the generates construction sequence as positive certificate G. Otherwise, decomposes into its 3-edge-connected components and each of them well bridges cactus representation cut-pairs in negative certificates. All these are done by making only one pass over using...