The -dominating cycles in graphs

For a graph G , let σ ̄ k + 3 ( G ) = min { d ( x 1 ) + d ( x 2 ) + ⋯ + d ( x k + 3 ) − | N ( x 1 ) ∩ N ( x 2 ) ∩ ⋯ ∩ N ( x k + 3 ) | ∣ x 1 , x 2 , … , x k + 3  are  k + 3  independent vertices in  G } . In [H. Li, On cycles in 3-connected graphs, Graphs Combin. 16 (2000) 319–335], H. Li proved that if G is a 3-connected graph of order n and σ ̄ 4 ( 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 σ ̄ k + 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.