On a kind of restricted edge connectivity of graphs Original Research Article

Let G=(V,E) be a connected graph and S⊂E. S is said to be a m-restricted edge cut (m-RC) if G−S is disconnected and each component contains at least m vertices. The m-restricted edge connectivity λ(m)(G) is the minimum size of all m-RCs in G. Based on the fact that λ(3)(G)⩽ξ3(G), where ξm(G)=min{ω(X): X⊂V,|X|=m and G[X] is connected} (ω(X) denotes the number of edges with one end vertex in X and the other in V⧹X), we call a graph G super-λ(3) if λ(m)(G)=ξm(G) (1⩽m⩽3). We proved that regular graphs with order more than 5 have at least one 3-RC, and show that vertex-and edge-transitive graphs other than cycles are super-λ(3). We also characterize super-λ(3) circulant graphs. As a consequence, we give the counting formula for the number of i-cutsets Ni of these graphs (including the Star graphs, the Hypercubes and the Harary graphs) for i, 2k−2⩽i<ξ3(G), where k is the regular degree of G.