科研之友
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摘要
For a connected graph G = (V, E), an edge set S subset of E(G) is called a k-restricted edge cut of G if G - S is disconnected and every component of G - S has at least k vertices. The k-restricted edge connectivity of G, denoted by lambda(k)(G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex,subsets X, Y of G, define [X, Y] = {xy is an element of E (G) : x is an element of X, y is an element of Y} and define xi(k)(G) = min{vertical bar[X, (X) over bar]vertical bar : X subset of V(G), vertical bar X vertical bar = k, G[X] is connected}, where (X) over bar = V(G) \ X. G is lambda(k)-optimal if lambda(k)(G) = xi(k)(G). Furthermore, G is super-At if every minimum k-restricted edge cut of G isolates a connected subgraph with order k. The k-restricted edge connectivity is an important index to estimate the reliability of networks. In this paper, some degree conditions for graphs to be maximally k-restricted edge connected and super k-restricted edge connected are given.
