Abstract
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 αl(G)=max{|S|:S⊆V(G),dG(x,y)≥lfor all distinctx,y∈S}, where dG(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 α2(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.
Published Version
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