The k-dominating cycles in graphs

For a graph G, let σ̄k+3(G) = min {d(x1) + d(x2) + · · · + d(xk+3) − |N(x1) ∩ N(x2) ∩ · · · ∩ N(xk+3)| | x1, x2, · · · , xk+3 are k + 3 independent vertices in G}. In [5], H. Li proved that if G is a 3-connected graph of order n and σ̄4(G) ≥ n+3, then G has a maximum cycle such that each component of G−C has at most one vertex. In this paper, we extend this result as follows. Let G be a (k + 2)-connected graph of order n. If σ̄k+3(G) ≥ n + k(k + 2), G has a cycle C such that each component of G− C has at most k vertices. Moreover, the lower bound is sharp.