Strongly Menger-edge-connectedness and strongly Menger-vertex-connectedness of regular networks.

Let F be a conditional faulty edge set of a graph G. Here the condition imposed on F is δ(G−F)≥r for some fixed integer r. The graph G is called F-strongly Menger-edge-connected of order r if each pair of distinct vertices u and v are connected by min{degG−F(u), degG−F(v)} edge-disjoint paths in G−F, where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. A graph G is t-strongly Menger-edge-connected of order r if G is F-strongly Menger-edge-connected of order r for every F⊂E(G) with |F|≤t. One can consider the vertex version in an analogous way.