m-dominating k-ended trees of graphs

Let k ≥ 2 , l ≥ 2 and m ≥ 0 be integers, and let G be a connected graph. If there exists a subgraph X of G such that for every vertex v of G , the distance between v and X is at most m , then we say that X m -dominates G . Define α l ( G ) = max { | S | : S ⊆ V ( G ) , d G ( x , y ) ≥ l for all distinct x , y ∈ S } , where d G ( x , y ) denotes the distance between x and y in G . We prove the following theorem and show that the condition is sharp. If α 2 ( m + 1 ) ( G ) ≤ k , then G has a tree that has at most k leaves and m -dominates G . This is a generalization of some related results.