Bounds for the Super Extra Edge Connectivity of Graphs

Let G be a connected graph, S be a subset of edges in G, and k be a positive integer. If G - S is disconnected and every component has at least k vertices, then S is a k-extra edge-cut of G. The k-extra edge-connectivity, denoted by lambda(k)(G), is the minimum cardinality over all k-extra edge-cuts of G. If lambda(k)(G) exists and at least one component of G - S contains exactly k vertices for any minimum k-extra edge-cut S, then G is super lambda(k). Moreover, when G is super-lambda(k), the persistence of G, denoted by rho(k)(G), is the maximum integer m for which G - F is still super-lambda(k) for any set F subset of E(G) with vertical bar F vertical bar <= m. It has been shown that the bounds of rho(k)(G) when k is an element of {1, 2}. This study shows the bounds of rho(k)(G) when k >= 3.