mm-dominating kk-trees of graphs

Let k≥2k≥2, l≥2l≥2, m≥0m≥0 and n≥1n≥1 be integers, and let GG be a connected graph. If there exists a subgraph HH of GG such that for every vertex vv of GG, the distance between vv and HH is at most mm, then we say that HH   mm-dominates GG. A tree whose maximum degree is at most kk is called a kk-tree. Define αl(G)=max{|S|:S⊆V(G),dG(x,y)≥lfor all distinctx,y∈S}, where dG(x,y)dG(x,y) denotes the distance between xx and yy in GG. We prove the following theorem and show that the condition is sharp. If an nn-connected graph GG satisfies α2(m+1)(G)≤(k−1)n+1α2(m+1)(G)≤(k−1)n+1, then GG has a kk-tree that mm-dominates GG. This theorem is a generalization of both a theorem of Neumann-Lara and Rivera-Campo on a spanning kk-tree in an nn-connected graph and a theorem of Broersma on an mm-dominating path in an nn-connected graph.