Restricted Edge Connectivity of Harary Graphs

Abstract

AbstractAn edge subset F of a connected graph G = (V,E) is a k-restricted edge cut if G − F is disconnected, and every component of G − F has at least k vertices. The k-restricted edge connectivity of G, denoted by λ k (G), is the cardinality of a minimum k-restricted edge cut. By the current studies on λ k , it can be seen that the larger λ k is, the more reliable the graph is. Hence one expects λ k to be as large as possible. A possible upper bound for λ k is ξ k defined as \(\xi_k(G)=\min\{\omega(S):\emptyset\neq S\subset V(G), |S|=k\ \)and\(\ G[S]\ \)is connected }, where ω(S) is the number of edges with one end in S and the other end in V(G) ∖ S, and G[S] is the subgraph of G induced by S. A graph G is called λ k -optimal if λ k (G) = ξ k (G). A natural question is whether there exists a graph G which is λ k -optimal for any k ≤ |V(G)|/2. In this paper, we show that except for two cases, the Harary graph has this property.