m-dominating k-ended trees of l-connected graphs

Let k≥2, l≥1 and m≥0 be integers, and let G be an l-connected graph. If there exists a subgraph X of G such that the distance between v and X is at most m for any v∈V(G), then we say that Xm-dominates G. A subset S of V(G) is said to be 2(m+1)-stable if the distance between each pair of distinct vertices in S is at least 2(m+1). In this paper, we prove that if G does not have a 2(m+1)-stable set of order at least k+l, then G has an m-dominating tree which has at most k leaves.