m-dominating k-trees of graphs

Let k (>=) 2, l >= 2, m >= 0 and n >= 1 be integers, and let G be a connected graph. If there exists a subgraph H of G such that for every vertex v of G, the distance between v and H is at most m, then we say that H m-dominates G. A tree whose maximum degree is at most k is called a k-tree. Define alpha(1)(G) = max{vertical bar S vertical bar : S subset of V(G), d(G)(x,y) >= l for all distinct x, y is an element of SI, 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 an n-connected graph G satisfies alpha(2(m+1)) (G) <= (k - 1)n+1, then G has a k-tree that m-dominates G. This theorem is a generalization of both a theorem of Neumann-Lara and Rivera-Campo on a spanning k-tree in an n-connected graph and a theorem of Broersma on an m-dominating path in an n-connected graph. (c) 2015 Elsevier B.V. All rights reserved.